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The Markovian Binary Tlee: A Model of
the Macroevolutionary Process
Nectarios Kontoleon
Thesis subm'itted
for the degree of
Doctor of Phi,losophA
Ln,
Appli,ed Mathemat'ics
at
The [Jntuerszty of Adelaide
(Faculty of Engi,neering, Computer and Mathemati,cal Sciences)
Discipline of Applied Mathematics
School of Mathematical Sciences
March L4,2006
Contents
Signed Statement
vll
Acknowledgements
vlll
Dedication
lx
Abstract
x
1 Introduction
1
2
1.1
Macroevolution and Mathematical Modelling
1
I.2
A guide to the thesis
4
Processes
2.7 Introduction
2.2 The Galton-Watson Process
2.2.L Defrnition
Branching
2.2.2
2.3
7
7
9
.....
Transience of the Non-zero States and the Extinction Probability 10
The One-Dimensional Continuous-Time Markovian Branching Process
11
2.3.I
Deflnition
11
2.3.2
Non-Explosiveness and the Mean of the Process
13
2.3.3
Transience of the Non-Zero States and the Extinction Proba-
bility
2.4
9
The Continuous-Time Markovian Multi-tvpe Branching Process
74
15
3
2.4.r
Definition
15
2.4.2
Non-Explosiveness and the Mean of the Process
19
2.4.3
TYansience of the Non-Zero States and the
Extinction Proba-
bility
20
Models of Macroevolution
22
3.1 Introduction
3.2 Phylogenetic Trees: Species Relationships
3.3 Tree Topology
3.4 A Labelling System for Binary Trees
3.5 Macroevolutionary Models
3.6 Probability Measures and TÏee Topology
22
22
.
23
.
3.6.1
27
31
32
Branch Types and Associated Probability Measures
3.7 Some Topological Concepts
3.8 Colless's Index of Imbalance
3.9 Birth and Death Model
3.
38
39
42
10 Proportional-to-Distinguishable-Arrangements Model
50
3.11 Multi-Rate Evolutionary Model
3.11.1 The PDA Model as an MR Model
,.t
Jt-)
59
.....
61
.
3.71.2 The super-PDA Model
64
4 Matrix Analytic Methods: an Introduction
4.7 Introduction
4.2 Phase-Type Renewal Processes
4.3 Markovian Arrival Processes
69
69
77
74
4.4
Level Independent Quasi-Birth-and-Death Processes
4.5
Level Independent Algorithms
83
4.5.L The Algorithm of Neuts
4.5.2 AlgorithmU...
83
4.5.3 The Level-Independent Logarithmic Reduction Algorithm
86
.
80
85
4.6
Level-Dependent Quasi-Birth-and-Death Processes
4.7
Level-Dependent Algorithms
89
4.7.7
89
88
.
The Level-Dependent Logarithmic Reduction Algorithm
g2
5 Markovian Binary Trees
5.1
,92
.96
.97
.97
.98
.99
Markovian Binary Tree: Definition
5.1.1 An Alternative
Representation of the States of the Process
5.2 An MBT is a special case of a cIMMTBP
5.2.7 Definition
5.2.2 Regularity and the Mean
5.2.3 Probability
5.3
5.4
Number of Branches
of Eventual Extinction
MBTs and Simple Macroevolutionary Models
.
100
5.3.1 Constant Rates Birth-and-Death Model
5.3.2 Proportional-to-Distinguishable Arrangements Model . .
5.3.3 The super-PDA model
.
100
.
101
.
104
The MBT and the Multi-Rate Model
.
110
5.4.7 The MBT Representation of the MR Model
5.4.2 The MBT-like Representation of the MR model
.
772
.
116
6 Probability Distribution of Imbalance
6.1 Introduction
6.2 The Imbalance Algorithm
6.3 Some Results for Simple Models
6.4
.
t24
. 724
. 725
.
729
6.3.1 The Constant Rates BD Model
6.3.2 The PDA Model
.
130
6.3.3 The sPDA Model
6.3.4 The Completely Unbalanced Model
6.3.5 A One Parameter Family of MBTs
1ÐO
. l.JJ
.
138
The Complexity of the Imbalance Algorithm
.
150
. t32
1Ð. IJf
7 Algorithmic Approaches for the MBT
7.I Introduction
7.2 An aside: Tree Labelling
7.3 The Depth Algorithm
t54
754
and Representation
155
L57
.
7.3.L A New Interpretation for the Sample Paths of the Neuts
AI-
gorithm
7.4
7.5
7.6
The Order of an MBT: Definition
169
The Order Algorithm
773
Comparing the Depth and Order Algorithms
7.6.7
7.7
8
I
762
180
.
Numerical Comparison of the Depth and Order Algorithms
Logarithmic Reduction Algorithms
The General Markovian TTee
182
.
184
188
8.1
Introduction
188
8.2
The Markovian T[ee: Definition
189
8.3
The Markovian Tree: cIMMTBP Representation
192
8.3.1 Definition
792
8.3.2 Regularity and Mean Number of Branches
8.3.3 Probability of Eventual Extinction
193
8.4
An Aside: Labelling the Nodes of an MT
195
8.5
The Depth Algorithm
797
8.6
The Order of an MT: Definition
200
8.7
The Order Algorithm
203
794
Conclusions and Further Research
2L4
9.1
9.2
Conclusions
274
Future work
277
Bibliography
2t9
List of Figures
A hypothetical phylogenetic tree
24
3.3.1 Two representations of the same tree
24
3.2.1
3.3.2
A tree with varying speciation rates
25
3.3.3 Two topologically isomorphic trees
26
3.4.1 An example of the evolution of an unstable leaf node.
27
3.4.2 Ãn example of the labelling of a binary tree.
29
3.6.1 Same number of branches at different times does not mean topology
is the same
35
3.6.2 Same number of branches at different times does not mean topology
36
is the same
3.7.1 The two topologically isomorphic classes of size four.
38
3.8.1 Colless's index of imbalance for two trees
40
3.10.
lA distinguishable arrangements example
4.3.1 Probability distribution
for an observable event against time given
thattheprocessbeganinphase2
4.5.1 Two sample paths in
51
...
52
.....
.....
5.4.1 An example of the pruning required for MR trees
5.4.2
^
79
84
111
72r
three branch topology
6.2.1 An illustration of the imbalance algorithm
. 728
6.3.1 The three topologically isomorphic classes of size 5
.
V
131
6.3.2 The completely unbalanced tree of size
6.3.3 Low imbalance
5..
138
.
r40
MBT model
6.3.4 Maximally imbalanced
r42
MBT model
6.3.5 Mean of Colless' index of imbalance for size 5 trees
t44
6.3.6 Magnification of the mean imbalance for small (.
745
6.3.7 One topology from each topologically isomorphic class of size 5.
746
6.3.8 Mean Imbalance for Size 6 Tlees
148
6.4.1 The computational complexity of the imbalance algorithm.
752
7.3.1 An example of an MBT of depth 5.
159
.4.I Ãn example of an order calculation
770
7
7.4.2Two different trees of order one.
t72
7.5.14n example of a U-unit.
775
7.5.2 Ãn example of an extinct tree
built from four [/-units
776
7.6.1 The space of trees included at the second iteration of the Depth al-
gorithm, with their order also indicated
180
.
7.6.2 A tree of order 1 that only appears at the 20-th iteration of the Depth
r82
algorithm.
7.6.3 Comparison of the Depth and Order algorithms as e varies from 0 to
0.5.
.
183
.
8.4.1Labelling nodes in an MT
.
196
MT of depth
.
198
8.5.1 An
5.
8.6.1 An example of an order calculation
. 20t
8.7.1 An example of a Ua-unit.
. 206
.
8.7.2 An example of a tree with three Lft-units.
. 206
Acknowledgements
I would like to thank the Federal Government of Australia for its financial support
through the Australian Postgraduate Award. Without this support I would not
have been able to undertake a PhD.
I would
also like to thank my two supervisors,
Professors Niget Bean and Peter Taylor. They have both been exceptional super-
visors always eager to help and always extremeiy supportive throughout my entire
PhD. I do not believe a student can ask for better supervisors. I would finally like
to thank my family who were also extremely supportive of my efforts during this
time. Thank you.
vlll
Dedication
This thesis is dedicated to my Dad. To teliosa Stelara mou! Thoxa to Theo!
IX
Abstract
One of the fundamental problems in biology is concerned with deciphering and un-
derstanding the nature of evolution. The results of evolution can be seen through
the diversity of life found on earth today. The relationships between species
can
be ascertained using a variety of biological and statistical techniques. These relationships can be pictorially represented on a tree diagram called a phylogenetic
tree. It
has been found that many phylogenetic trees are imbalanced, meaning
that the subtrees of phylogenetic trees differ in shape. The focus of this thesis
is
to develop physically-reasonable mathematically-tractable models of the speciation
process. We do not wish to model the evolutionary process at the genetic level but
rather, to model the process at the species level as represented by the branching
structures of phylogenetic trees.
The simplest models of macroevolution generate branching structures that are
either too balanced or too imbalanced. Therefore
it
has become increasingly fash-
ionable to model the macroevolutionary process using continuous-time Markovian
multitype branching processes (cIMMTBP). Continuous-time MMTBPs provide the
flexibility needed to generate tree structures with any level of imbalance. However,
the major pitfall of using ctMMTBPs is that they do not have an algorithmic approach for ascertaining measures useful in macroevolution.
The model that is proposed is called the Markovian binary tree and provides
an alternative representation of the binary-split ctMMTBP. This representation is
made possible by re-interpreting the transition structure of the cIMMTBP. The
MBT has sufficient flexibility to account for the variation in branching structures
X
of phylogenetic trees and is amenable to algorithmic analyses. MBTs can also be
written
as level-dependent quasi-birth-and-death processes
(LDQBD).
We show that many of the current models of the macroevolutionary process
are subsumed by the MBT. In particular, we show that the most flexible of these
models, the Multi-rate model (MR), which is also a cIMMTBP, can be subsumed by
the MBT in the limit as ú ---+ oo. We do this by transforming the MR into an MBT.
This model has a simpler interpretation than the MR model and now the probability
that
a random tree eventually evolves
to some topology, T , has an analytic solution.
Since the MBT is a LDQBD, the myriad numerical algorithms within the theory
of matrix analytic methods can be modified to apply to the MBT. Indeed, we show
that despite the MBT being a level-dependent QBD process, two level-independent
algorithms can be modified for determining the probability of eventual extinction
of the process. These algorithms are called the Depth and Order algorithms and
are based on different physical interpretations of the evolution of
MBTs. These
algorithms can also be applied to find the extinction probability of MR model trees.
Surprisingly, we show that level-independent quadratically convergent algorithms
cannot be modifled to the MBT and that level-dependent quadratically convergent
algorithms are generally less efficient than the lineariy-convergent Order algorithm.
We also develop an algorithm for the MBT that determines the average imbalance.
The MBT is generalizedto the Markovian tree (MT), characterized by the fact
that branch points need not be binary. The MT provides an alternative framework
for the ctMMTBP and bridges the gap between branching processes and matrix
analytic methods. Finaliy, we provide the Depth and Order algorithms for the MT
modei.
Chapter 1
Introduction
1.1 Macroevolution and Mathematical Modelling
Earth is home to a staggering amount of diversity of life. How did such diversity
arise? The difficulty in answering such a question makes
attempt to solve
it.
it
all the more enticing to
One can begin to understand the mechanisms behind evolution
by studying the process at the microscopic level, that is, by studying the
changes
that occur at the genetic level, or at the macroscopic level, that is, by studying
the changes that occur at the species level. In macroevolution to be more specific,
we are concerned with identifying the differences between species, quantifying these
differences and then understanding just how and why these differences arose. The
relationships between species can be represented pictorially in diagrams called phylogenetic trees. Phylogenetic trees give information on how related two species may
be and in some cases predict the time since these species diverged from their most
recent common ancestor.
There are, of course many problems associated with the biological and statistical
determination of phylogenetic trees 122]. For example, one very important source of
information, the paleaontological record (the fossil record) is incomplete. Therefore,
in order to infer the phvlogenetic tree shape from an incomplete data set requires the
1
CHAPTER
1, INTRODUCTION
2
use of statistics and a stochastic model of the macroevolutionary process. This can
in principle then, produce a phylogenetic tree shape that has the highest probability
of representing the actual tree shape [22].
Phyiogenetic tree shape is important [1, 10, 11, 12, 15, 22, 30, 26, 33,34] because
it
gives clues as to how the rates of macroevolution,
that is, the rates of species
generation and the rates of species extinction, have changed over time and in different physical locations
1221.
The rate of change of the macroevolutionary process
can have profound effects on the shape of the phylogenetic trees [22]. Phylogenetic
trees that demonstrate significant rate variation are imbalanced. That is, differ-
ent portions of the tree have different shapes. For example, some portions of the
tree may be densely populated with many short branches, whereas other portions
may be sparsely populated with long branches. Therefore the shape of well constructed phylogenetic trees can give clues as to the processes that may have driven
macroevolution and thus generated tree shape [22].
As we have stated above, to aid in the construction of phylogenetic trees one
needs to make use of stochastic modelling [1, 11,
22,26,30]. In order for a model
to be reasonable, it must have the ability to generate useful information and to
be
mathematically tractable with physically reasonable assumptions.
Stochastic models are important in that they provide a probability distribution
over the finite number of possible phylogeneiic tree shapes that have a finite number
of species. The stochastic models that have been utilized [10, 11, 30] are very simple
in that they do not allow for any variation in the rate of the macroevolutionary
process. One of these simple models is the well known, constant-rates birth-anddeath (crBD) model and another is the proportional-to-distinguishable arrangements
(PDA) model,
see 122]
and references therein. As expected, these models cannot
account for the levels of imbalance that are found in phylogenetic trees, because they
do not allow for rate variation. The crBD model predicts trees that are too balanced
whereas the PDA model predicts trees
that are too imbalanced [30]. Consequently,
the next step in the development of physically reasonable mathematically tractable
CHAPTER
1. INTRODUCTION
3
models is to allow for rate variation.
The process of macroevolution can be thought of as a continuous-time branching
process. That is, a process that begins with some particles that have the ability to
generate new particles at random time intervals. This is exactly what is happening
at the species level in macroevolution, a species will at some random points spawn
a new species.
is generated
It
1221.
is generally believed that at any time point only one new species
This is a reasonable assumption, since
it
seems
unlikely that two
or more new species will be created simultaneously. The use of more sophisticated
branching process models was originally suggested by Mooers and Heard l22l and
then re-iterated by Aldous [1].
The continuous-time Markovian multi-type branching pïocess (ctMMTBP)
21] is an excellent candidate for a macroevolutionary model because
it
[2,
allows for
variations in the rates of speciation and variations in extinction rates. Unfortunately
though, the cIMMTPB is difficult to analyse and there is very little algorithmic
development.
Despite this, Pinelis [26] proposed a model based on the continuous-time Marko-
vian multi-type branching process (cIMMTBP) catled the multi-rate (MR) model.
It
was called the multi-rate model to emphasize the fact
that this model allows for
significant rate variation. The MR model assigns to each species individual specia-
tion and extinction rates. For example, some species have the capacity to generate
new species more rapidly than others, whereas other species can become long-lived
evolving only very slowly. The MR model encompasses all the models that do not
allow for rate variation.
In this thesis we propose a model of the macroevolutionary process that is
also
a continuous-time Markovian multi-type branching process which we have called
the Markovian binary tree model (MBT). The MBT requires us to interpret the
cIMMTBP in a subtly different way, This new interpretation admits a different representation to the conventional cIMMTBP representation. Consequently, a whole
new vista of modelling flexibility is opened up to the MBT because this representa-
CHAPTER
1. INTRODUCTION
4
tion provides an excellent platform from which to develop a sound algorithmic basis.
Consequently, the answers to questions that a biologist may have can be potentially
solved using the MBT. Thus, due to its representation and interpretation, the MBT
has a significant advantage over the MR. In fact, the MR model can be shown to be
encompassed by the MBT.
In the next section we discuss the layout of this thesis.
L.2 A guide to the thesis
We begin by giving an introduction to the world of branching processes in Chapter 2.
Branching processes have a rich history of theoretical development 12,91. The first
process that we discuss is the discrete-time Galton-ril/atson process, the cornerstone
of branching process theory. This process is then generalized to its continuous-time
counterpart, the continuous-time Markovian branching process. Following these
preliminaries we then discuss the continuous-time Markovian multi-type branching
process. This branching process provides the core from which the MBT is constructed and we therefore take some time in explaining
In
it
carefully.
Chapter 3 we begin by discussing the macroevolutionary biological back-
ground. We briefly introduce phylogenetic trees and then discuss some of the im-
portant tree topological concepts. The next step we take is to discuss the most
important quantitative measure of tree imbalance: Colless' index of imbalance.
The remainder of Chapter 3 is devoted to introducing some of the most important
macroevolutionary models. The constant-rates birth-and-death (crBD) model which
has an important place in applied probability, the proportional-to-distiguishable ar-
rangements (PDA) model, the super-PDA model and finally the multi-rate (MR)
model of Pinelis [26]. We also show how the crBD model generates the PDA model.
Finally, a discussion of the MR model is given.
Having introduced branching processes and the biological background we next
introduce the theory of matrix analytic methods in Chapter 4. We commence by dis-
CHAPTER
1. INTRODUCTION
tr
ü
cussing the Poisson process, followed by the phase-type renewal process. The phase-
type renewal process is the generalization of the Poisson process to non-exponential
inter-event distributions. We next discuss the Markovian arrival process (MAP).
The MAP is the generalization of the phase-type renewal process to include correla-
tions. The MAP generates the dynamics of the MBT. The concept of the hidden and
observable transitions of the MAP is used to alter the interpretations of particle tran-
sitions in the ctMMTBP and create the MBT interpretation. The level-independent
quasi-birth-and-death process (LIQBD) is then introduced and we analyze the al-
gorithm of Neuts, the algorithm U and the level-independent logarithmic reduction
algorithm. The algorithm of Neuts and the algorithm U form the basis for analogous algorithms for the MBT that determine the probability of eventual extinction.
The flnal process we discuss is the level-dependent quasi-birth-and-death
process
(LDQBD). The LDQBD process is the framework within which we represent the
Markovian binary tree. The last topic we discuss is the level-dependent logarithmic
reduction algorithm.
In Chapter 5 we begin by representing the Markovian binary tree (MBT)
as a
level-dependent quasi-birth-and-death process. We re-interpret the cIMMTBP process such
that each evolving branch of an MBT has its own copy of the MAP.
Since
the MBT is a cIMMTBP, more speciflcally, a binary-branch point ctMMTBP,
we
also write the basic branching process equations for the MBT. From these equations
we obtain the equation for the probability of eventual extinction of the process. The
final sections of Chapter 5 are devoted to showing that all the models discussed in
Chapter 3 are special cases of the MBT. We show, in particular, that the MR model
can also be written in terms of an MBT and is thus subsumed by the MBT.
In Chapter 6 we demonstrate the power and flexibility of the MBT by developing
an algorithm that calculates the mean imbalance conditional on tree size. We show
that there exists a simple MBT with one parameter that has sufficient flexibility to
span the entire range of theoretically allowed imbalance values for size five trees. \Me
also demonstrate that even though this one parameter model was designed specif-
CHAPTER
1. INTRODUCTION
6
ically for size five trees, this model stiil generates interesting behaviour for larger
size trees.
It still spans most of the allowed imbalance
values and therefore retains
much of the flexibility seen for size 5 trees. The final section of Chapter 6 is devoted
to calculating the computational complexity of the algorithm.
Chapter 7 continues the algorithmic development of the MBT, where we specif-
ically concentrate on finding the minimal non-negative solution to the equation for
the probability of eventual extinction of the MBT process. We begin by developing
the Depth algorithm which is analogous to the algorithm of Neuts. We show that
the difficulty in describing the sample paths in the algorithm of Neuts is removed if
the sample paths are transformed into binary trees. The Order algorithm which is
analogous to the algorithm
[/ is also developed. This algorithm has an interesting
physical interpretation based on a concept called the order of a tree. The Order
algorithm is shown to converge linearly with respect to order. A comparison of the
Depth and Order algorithms is made and we show that the Order algorithm converges at a faster rate than the Depth algorithm because
it
considers more topologies
at each iteration. We conclude Chapter 7 by analyzing the quadratically convergent
logarithmic reduction algorithms.
It
is shown that a level-independent logarithmic
reduction algorithm is not possible for the MBT and that the level-dependent loga-
rithmic reduction algorithm will in general perform worse than the Order algorithm.
The success with which algorithms were deveioped in Chapters 6 and 7 leads us
to the generalization of the MBT. The general Markovian tree (MT) is introduced
in Chapter 8. We begin by representing the MT in a matrix analytic form, just as
we did for the MBT, and then write the general cIMMTBP definition of the MT. By
writing the general ctMMTBP as an MT we commence developing algorithms that
may be of use in a physical modelling context. Therefore as a starting point, we
develop the Depth and Order algorithms for the probability of eventual extinction
of the MT. These algorithms reduce to the Depth and Order algorithms of the MBT
if
each branch point is forced to be binary.
Chapter
2
Branching Processes
2.L Introduction
Evolutionary biologists face the daunting task of providing a framework with which
to explain the observed diversity of life found on earth. The relationships between
the species can be represented through the use of tree diagrams, called phylogenetic
trees. The task then, is to decipher the shape of the phylogenetic tree of life and
to determine the mechanisms that generated that particular shape. However, given
the incompleteness of the biological record and the scarce knowledge of the factors
that cause macroevolution, this is indeed a daunting task. At a more modest level,
evolutionary biologists have studied the shapes of some of the subtrees of the tree
of life by using biological and statistical techniques. As a result, there is now an
emphasis on developing models of the macroevolutionary process [1, 11, 70,22,26,
301.
There are two possible avenues with which to pursue the development of a model
of macroevolution,
o to develop a model that is based soiely on physical considerations, or
o to construct a model that can account for the tree shapes that arise in nature, without attempting to provide a complete mechanistic basis for their
7
CHAPTER
2.
BRANCHI¡úG PROCESSES
8
generation.
The first approach is currently extremely difficult to implement since
it
is plagued
by a lack of understanding of the underlying biological mechanisms that cause
macroevolution. In this thesis, we choose the second approach. Thus we shall
develop a model that can account for the tree shapes that appear in nature, which
in addition is also based on some reasonable physical considerations.
Qualitatively then, a species under some evolutionary constraints will continue
evolving and at some point during its evolution will either become extinct or give
rise to new daughter species while
it then continues to evolve. Viewed in this light,
a
branching process seems to be a perfect candidate as a model of macroevolution. In
fact, Mooers and Heard [22] stated this very succinctly, "rnost biologi,cal tara
arisen by a branching process of descent wi'th modi,ficati,on)'
haue
.
The remainder of this chapter is devoted to discussing some important branching
processes. The aim is to describe the fundamental nature of the models currently
used
in macroevolutionary modelling in addition to allowing us to introduce
the
model that is proposed in this thesis. This chapter is organised as follows. In Sec-
lion 2.2 we discuss the simplest type of branching process called the Galton-Watson
process (GW). The Galton-Watson process is a discrete-time single-type branching
process and is the simplest of al1 the branching processes. In Section 2.3 we describe
the continuous-time analogue of the GrrV process: the single-type continuous-time
Markovian branching process (ctMBP). Finally, in Section 2.4 the ctMBP is generalized to the multi-type analogue, called the continuous-time Markovian multi-type
branching process (cIMMTBP).
CHAPTER
2.2
2.
I
BRA¡\ICHI¡\IG PROCESSES
The Galton-'Watson Process
2.2.L Definition
Excellent introductions to the Galton-Watson process can be found
in [2] and [9].
Let the random variable Z¿ denote the number of particles that are present at time
I given that the process commenced with one particle at time 0. Each particle that
is present evolves independently of all the others and of its preceding history. At
time I * 1 a particle can either give rise to no offspring with probability p6 or with
probability p¡" give rise to k daughter particles, for k
)
1.
The generating function of the offspring distribution of one particle is given by,
oo
/(")
: wlt",): t
pksk,lsl> I,
(2.2.1)
k:0
and the expected number of particles in the first generation spawned by a single
particle is given by
E(zt):+l : îror
ts:l
(2.2.2)
k=O
The iterates of the above probability generating function are,
"fo(t)
:
",
/t(")
: /(t),
/"*r(t)
: lU"G))'
(2.2.3)
Let P(i,,,m)be the one-step probability that the process will have m particles given
that there were
i
at the previous step, in other words,
P(i,m)
: PIZ+,: ml Z¿: i']
(2.2.4)
Clearly then,
I
t(t, m)s-:
rn:o
Suppose that the process commences w
/(").
th a particles and
(2.2.5)
since the offspring dis-
tribution at the next generation is the sum of e independent random variables, the
CHAPTER
2.
BRA¡\ICHI¡\IG PROCESSES
10
probability generating function for the offspring distribution is given by the convolution of the z individual offspring probability generating functions.
Hence,
oo
,(u,m)s^ :
Ð
m,:o
(2.2.6)
[/(s)]',
for i, ) I.
Let P,(i,m) be the probability that there are m particles at
nl1
given that
there were'i particles at time 1. Then, following, [2],
oo
Ë **,(t ,m)'^
m:0
oo
t t
m:O
Pn(r,k)P(k,m)s^
(2.2.7)
l<:O
oo
oo
¿(t, ÐD, P(k,m)s*
Ð
m:o
k:o
(2.2.8)
oo
e(t, r;[/(")1*,
D
k:0
(2.2.e)
where in the frrst step we have used the Chapman-Kolmogorov equations, and in
the final step we have used equation (2.2.6).
Using arguments similar to equation (2.2.6)
it
can be shown that,
ôo
D r,(n,m)s^: [/"(r)]o
(2.2.10)
rn:o
In other words, the n-th step probability generating function of the process is given
by the product of the n-th step probability generating functions of each of the
i,ndi,ui,dual branching processes commenced
2.2.2
by the
ri
initial particles.
Transience of the Non.zero States and the Extinction
Probability
It
has been shown [9] that all the non-zero flnite particle states of the process are
transient, thus, with probability orre, Z¿ ---+ 0 or
Zt-
æ as I --+ oo
The probability of eventual extinction, q, of the process is the probability that
I ---+ oo there
are no
living particles remaining. In other wotds, g : lim¿--
as
PlZr:91.
CHAPTER
2.
BRA¡\ICI{ING PROCESSES
It can be shown, [2, 9] that the extinction
11
probability is the smallest non-negative
root of the equation
r:
"f
(").
(2'2'11)
it can also be shown that, if E(Zt) I 1 and the variance is greater
than zero when E(Zr) : 1 then Q : 7 and if E(Zt) > 1 then q < l. The process is
called subcritical ff E(Zr) < 1, critical if tr(21): 1 and supercritical tf E"(21) > l.
Furthermore,
Now, since all the non-zero finite particle states of the process are transient we have,
Pt,JlT
2.3
Z¿
: 0l: q :1 - Pt,llt Zt: æl
(2.2.12)
The One-Dimensional Continuous-Time Marko-
vian Branching Process
2.3.L Definitron
Consider the following continuous-time process: a particle that is alive at time
will live for an exponentially distributed lifetime with mean
7f
a,
al,
which point
it
)
7
will either give rise to no offspring with probability po or will give rise to m
offsping with probability
p-.
ú
Each particle evolves independently of all the other
particles and of its history. Such a process is called a one-dimensional continuous-
time Markovian branching process (ctMBP). The probability generating function of
the offspring distribution is again given by,
/(') : Do,,'^
(2.3. 1)
m:O
Now let
Z(t)
mlZçO¡:
denore the number of particles alive at time
1] be the probability that at time ú there a,re
process commenced
t. Let Pt,"(t) : PIZ(t) :
rrù
particles given that the
with only one particle. The probability generating function of
CHAPTER
2.
BRA¡úCI{I¡úG PROCESSES
the number of particles at time
12
ú is,
oo
F(s,t): t P6(t)s^.
(2.3.2)
m:0
* r given that it
PIZ(ttr) : mlZçr¡:11.
hadz particles at time r is denotedby n^Qir;r):
Now the process is homogeneous with respect to time, so fl^(t -f r;r) : P,*(t).
The probability that the process will have m parLicles by time
Since each particle evolves independently
ú
with respect to all other particles, the
probability generating function for the number of particles given i initial particles
is the ø-fold convolution of ,F(s,ú), hence,
æ
oo
\
;-r
eo^çt1t^
: l\/ e,,,{t¡t^ \'| :
\m:o
/
[tr'1s,
r)]'.
(2.3.3)
Let Q¿¡ be the rate at which the process goes from a state with i particles to
state with
j
particles. We then have
Qoi: iaqi-¿+r,
Tor
j:i'-7
a
a:nd
j > i', Qoj:0
for
j <i-
(2.3.4)
1, andfinally lor
i': j
ôo
Q¿¡ :
-'itpo - i* D Pt ¡+t:
-i'c,(I
- p),
(2.3.5)
le:i-17
since
fp
opk
:
1. The interpretation of Qq is as follows. The rate at which
single particle spa\Mns
j -i,+
1 particles can be seen
each
to be ap¡-¿¡l since the particle
must die, which occurs with rate a, and at its death
it
spawns
j - i,+ 1 particles
with probabiliíy p¡-¡¡1. Since any one of the 'd particles can do this, the total rate
is therefore i,apj-¿+t. Thus the total number of particles is
initial particles and the
j - i,+ 1 newly spawned
j,
comprised of the
z
-
1
particles.
We can now write down both the Kolmogorov forward and backward equations
for this system. The forward equation is,
d,
Ôo
:t P,t-(t)Qr¡,
än,(¿)
k:t
(2.3.6)
CHAPTER
Since
2.
BRA¡úCfTI¡üG PROCESSES
Qrj:0for j <k-
13
lwehave,
*r,,(r)
:
Ð
(2.3.7)
P¿n(t)eu¡
substituting equations (2.3.a) and (2.3.5) into equation (2.3.7) we obtain,
llu
j+l
/ :-
p..(t\
xJ \"
dtt
(.t)
n p,,xJ'
J
-,; "'
+
t*:,
pi¡,(t)kap¡-¡"¡1
(2.3.8)
The backward equation can be derived from
Å
ftntø: t Qo*P¿n(t),
(2.3.e)
and yields
Ôo
,'t
u p..(t\
dt'
pn-¿+rPn¡(t).
zJ\") - -inp.(¿) + io D
(2.3.10)
k:i_l
By multiplying equations (2.3.8) and (2.3.10) by s' and then summing from
i :0
to infinity we get,
*rrr,t)
: u(s)ftrçr,t¡, (forward equation),
(2.3.11)
and
*rrr,t)
: u(F(s,t)).,
(backward
equation),
(2.3.12)
where
z(s)
:"(/(r) -")
(2.3 13)
2.3.2 Non-Explosiveness and the Mean of the Process
In Harris, [9], it was shown that the process is non-explosive, that is, Z(t) < oo for
all ú < oo almost surely, if
: N,
[' ,r4,t
Jr-,fG)-s-*'
(2.2.14)
2.
CHAPTER
for every
e
)
BRA¡\ICIíD\rG PROCESSES
74
0. The condition
4
r(r)l . -,
ls:r
(2.3.15)
d,sr
is sufficient to ensure that the process is non-explosive [2]. This condition implies
that the mean number of particles produced by a single particle upon its death
is
flnite.
The mean number of particles of the process at time ú is defined by,
M(t) :Elz(t)lz(0)
Since,
: rl.
M(t) : *¿F'(r, ¿)1":, we can differentiate the Kolmogorov
(2.3.16)
backward equation
with respect to s to obtain,
4*rtl:
dt
^M(t),
where
Recall thaf
*f
G)
À: ftuþ,I":, : - (*rurl":, - ,)
l":, i. just
(2.r.rr)
(2 3 18)
the mean number of offspring generated when a particle
expires. The solution to equation (2.3.17) is given by,
M(t): exp(Àú),
and observe that,
(2.3.19)
if
o
À
)
0 then lim¿-oo
M(t):
o
À
:
0 then lim¿-oo
M(t) :1 and the process is critical, in fact M(t) :1¡ot
oo and the process is supercritical,
all ú, and finally if,
o
À
(
0 then lim¿-oo M (t)
2.3.3 TYansience
:
0 and the process is subcritical.
of the Non-Zero States and the Extinction
Probability
Harris [9] has shown that the Galton-Watson process is imbedded in the contintroustime Markovian branching process. Since all the non-zero finite-particle states of the
CHAPTER
2.
BRA¡üCIil¡\rG PROCESSES
15
Galton-Watson process are transient, this implies that all the non-zero finite-particle
states of the ctMBP are also transient and as a result,
Pt,ll1 z(t):01
: t - Pt,ll* z(t): æl
(2.3.20)
Define the probability of extinction at time ú to be
q(t)
:
- 1l :
Plz(t) :0lz(0)
F'(0,f)
It is not hard to see from the deflnition of F'(s, ú) that
(2.3.2t)
q(ú) is a non-decreasing
function of ú. From the Kolmogorov backward equation (2.3.10) one obtains,
d,
rtø(t)
with initial condition q(0)
:
: u(q(t)),
0. It is shown in
[2]
that q : lim¿-*
(2.3.22)
q(ú) is the minimal
non-negative solution of
u(s)
:9.
(2.3.23)
2.4 The Continuous-Time Markovian Multi-type
Branching Process
2.4.L Definition
Having discussed the Galton-Watson and the one dimensional continuous-time Marko-
vian branching process we are now in a position to introduce the continuous-time
muiti-type Markovian branching process (ctMMTBP). The main point of difference
between this process and the one dimensional process, is that there are now n dif-
ferent particles types as opposed to only one type in the ctMBP'
We shall follow the development in Athreya and Ney [2]. Suppose we have a
with n-particle types, each particle of type i' e {7,. . . ,n} has a life-span
that is exponentially distributed with mean 7f a¿, atd upon its death will produce
offspring of the n-types with distribution pØ(ir,i2,...,J",), where in € {0}UV'+
process
2,
CHAPTER
BRA¡üCTII¡{G PROCESSES
16
represents the number of particles of type k € {1, . . . ,n} that will be spawned. The
particles upon their birth evolve independently of each other and of the past.
The offspring probability generating function given that the process begins with
one particle of type 'i, for i,
¡(r)(sr, s2t...,s",)
e {I,2,.
:
. . , n}, ts
p(n)(ir,i2,...,i,-)tir"ti ...st;.
t
jt'i2,...,in>o
(2.4.L)
We say that the pïocess is singular if the generating functions (2.4.1) only consist of
for aIIi, e {1,...,n}. We call a branch point a singular
branch point, if an i,-type particle transforms into a i Type particle, for i' I j. A
terms that are linear in
s¿
binary branch point occurs when a particle of type
daughter particles of types
j
and k, for any i,k:
i
terminates and spawns two
L,2,...,h.
j : (ju..., j,) and i : (h,...,in) denote two vectors such that j¡,,i¡, e
{0} u Z+ for all k e {1,...,r}. LeT Zi(t) : (Zi(t),...,7i,QD be the number of
particles of each type at time ú given that the process began with i particles at time
0. Let P(i, j;ú) be the probability that a process beginning with i particles at time
0 will have j particles at time ú. The generating function is given by,
Let
F(i,s;t) :Elszrl'(t)
.
..
sz:{(t)l:
i
where s
:
D
P(ó,i;t)rir'
.
..rt;,
e(o\oz+¡^
("r, ... , s,) and ({0} UZ+)" is the r¿-fold cartesian product of {0} UZ+.
(t)
ttl
For ease of exposition we henceforlh denote szrl' . . . ,tj'" ") by 6zi
Lel e¿ be the vector with one in the z-th component and zero
n
-
(2.4.2)
7 components. Let
Zi(t) be the number of particles of type i
in the other
present at time
ú. Due to the independence of the evolution of each of these particles, each particle
initiates another multi-type branching process. Therefore, Ief Z!'i (r) be the number
of particles of type
of length
¡.
j
Thaf are generated by the
k-th particle of type z in a time interval
The total number of particles of type
is given by the sum of the type
j
j
that are present at time t
I r
particles generated by the Z¿(t) particles at time
2.
CHAPTER
ú
for all
,i
:
77
BRA¡úCHD\rG PROCESSES
7,2,.
..
,n. As a result, the total number of particles of type j
n z¿(t)
zj(t+")
are
:tÐt!"(r)
t:l
(2.4.3)
le:l
In terms of the generating functions of the particle distribution, equation (2.4.3) can
be written as
F(s;t I r) :,F(.F(s; r);t),
where .F(s;ú)
(2.4'4)
: (F("0,s;t),..., F(en,s;f)) and
F ("0,s; ú)
:
Efsz't'
(t)1'
The Kolmogorov differential equations play an important role in the theory of
Markovian processes. For the case of the cIMMTBP both the forward and backward
equations were first derived by Sevastyanov [32]. The forward equations are)
!rGr,s;ú)
Ot
: irt*'t")3r("
o,s;t),
oSt.
(2.4.5)
-
where
,(r)(s)
for all z e {1, 2,.
. . ,,n}.
:
ao¡¡u')(rr,..., s",) -
s¿],
(2.4'6)
Sevastyanov [32] cleverly derived the Kolmogorov forward
equations using a probability generating function approach.
The backward equations are given by,
â
ärþr,s;ú)
for all k e
{7,...,n}.
: u(k)[r(s;t)],
(2'4'7)
The backward equations have a simple physical interpre-
tation, and consequently they can be derived in a more intuitive fashion than the
forward equations. We shall use the argument as presented in [5]. Let the process
commence
with one particle of type k. The lifetime of this particle, 7, is exponen-
tially distributed with parameter, a¿, thus Pn(T S t)
: I-
exp(-a¿ú). Now by
conditioning on the lifetime of the particle we have'
F
("r,
s;
t)
:
Ml"t"
*(¿)
|
? > úl exp( - a¿, .
l:
F,lsz"n
(t)
lT
:
rla¡
exp(- a¡,r) d,r.
(2.4.s)
CHAPTER
2.
BRA^ICI{I¡\IG PROCESSES
18
The first term in equation (2.4.8) represents the situation where the original particle
has not yet died, as result
F-1"""*
it
is the only particle in the process, and therefore
(Ð1" > f]exp(-a¿t)
:
(2.4.e)
tnexp(-a¿ú)
The second term of equation (2.4.8) represents the situation where at time T
ú,
j
the initial particle dies and generates
each of the
j
: rI
new particles. In the remaining Time t
- r,
new particles (spawned from the original k-type particle) generate
their own cIMMTBP. Thus,
Elsz"n{t)lf
:
r]
Ipr*l
Ø)F("t, s;t
- ,)i' . . .F("n, s;t - r)i^
J
¡{t)(ra(s;
t-r))
(2.4.10)
Substituting Q.a.9) and (2.4.10) into equation (2.4.8) we obtain
F("^,s;ú)
:
s¿
exp(-ø¿ú) +
l"' ¡{t)
(r(s; t - r))a¡,exp(-a¡,r).
(2.4.11)
If we multiply through by exp(ø¿ú), we obtain,
F("r,
s;ú) exp(a¿t)
: t* +
t: ¡{t) (r(s; t - "))
Now changing the variable of integration from
F("r,
s;ú) exp(ø¿t)
:
tn +
1",
r
a¡, exp(a¡,(t
to u - t
-
- r))dr.
(2.4.L2)
ø we obtain,
¡{t)(r(s; "))orexp(a¡,u))d,u.
(2.4.13)
Finally differentiating equatíon (2.4.L3) with respect to ú, using the Fundamental
Theorem of Calculus and then multiplying through by exp(-ø¿ú) we obtain the
Kolomogorov backward equation,
ã
h,r@r,.s;¿)
forall ke{7,...,n}
: z{k) [r(s;t)],
(2.4.t4)
CHAPTER
2.4.2
2.
BRA¡\ICT{I¡\IG PROCESSES
19
Non-Explosiveness and the Mean of the Process
The process is not explosive, that is, regular, [2]
ô/(¿)is;
I
arîI":"
for all
if
_ ^^
' -'
(2'4'15)
j : 7,2,...,rù, where e is a vector of ones of the appropriate
'i,
dimension
and two vectors are considered equal to each other if all their components are equal.
In other words, the process is non-explosive if the expected number of particles
of any type j given that a birth occurs from a particle of type i is finite for all
i,j:7,2,...,n.
The condiLion (2.4.75) can also be shown [2] to imply that
m¿¡(t):ElZj(t)lz(o):
",1
.*
(2.4.t6)
Let the matrix of the expected number of particle types at time ú be denoted by
M(t) : {*o¡(t)l¿, j :
1, . .
., n}. From equation (2.4.4)
it
is easy to show that M(t)
satisfles the semi-group property [2], namely
M(t+u):
lor
t,u )
M(t)M(r),
0, and from equation (2.4.14) to show the continuity condition,
líryM(t)
where
l
(2.4.17)
: I,
(2'4'18)
is the r¿xr¿ identity matrix. Now (2.4.17) and (2.4.L8) imply that there exists
a matrix
A
[2] which is the infinitesimal generator of the semigroup
{M(t)l¿ >
O}
such that
M(t): exp(Aú).
Each element of the matrix
A, say A¿¡, can be interpreted
at which a particle of type 'l gives rise to particles of type
(2.4.1s)
as being the average rate
j.
In other words,
A¿¡ is
given by the rate, a¡ at which a particle of type z gives birth multiplied by the mean
CHAPTER
2.
BRAAICI{I¡úG PROCESSES
number of particles of type
we write,
A¿j:
a¿b¿¡
j
20
that are a created by that initial type i particle. Thus
where
^ --al'Ð(")
"ut,
ar¡ |l* ="--uu,,
(2.4.20)
"r¡
where
7
tf.
i,: j
0
if
i+ j
if
there exists some ú
The process is called positive regular
>
0 for all
(2.4.21)
:
to, such that
j.
Flom the theory of positive matrices [31] there exists a
strictly positive eigenvalue p(t6) of M(to), called the Perron-Flobenius eigenvalue,
m¿¡(to)
i,,
whichhasthepropertythatanyothereigenvalue pof M(to) issuchthatlpl <p(to)
and the algebraic and geometric multiplicities of the Perron Fþobenius eigenvalue
fori : I,2,. . . ,n,
where for all'i, À¡ are the eigenvalues of the matrix A. Both M(t) and A have the
same eigenvectors. Now let À1 be such that p(to): exp(À1ú6). Consequently, À1 is
are both one. The eigenvalues of
M(t)
are of the form, exp()¿ú)
real and )1 > Re(Ài,) for all À¿ :2,3,...,n, [31].
2.4.3 Transience
of the Non-Zero States and the Extinction
Probability
The proof of the transience of the non-zero finite states and of the extinction proba-
bility
are well known and relatively simple for the discrete-time
multi-type branching
process (also known as the Galton-Watson multi-type branching process) [9]. With
minor modiflcations these proofs carry over to the continuous-time Markovian multi-
type branching process [2, 9]. Thus,
if the continuous-time
Markovian multi-type
branching process is positive regular and non-singular, all states with a finite number
of particles are transient. Hence with probability one all realizations of the process
will either eventually become extinct or the total number of branches will tend to
infinity
[2].
CHAPTER
2,
2l
BRANCHI¡úG PROCESSES
Let q@ be the probability that the process beginning with one particle of type
z
will eventually become extinct. Let q:
(q(1),
...,q(")). It
can be shown [2] using
the backward equation (2.4.14) that q is the minimal non-negative solution of
z(s)
where
:
g,
(2.4.22)
ø is given by equation Q.a.6). This is equivalent to the condition
[9]
q: l(q),
where
/
(2.4.23)
is given by equation (2.4.7), for the discrete-time case. In fact, one can
show, as Harris [9] did for the discrete-time case, that the solution q is either equal
to e, or all its components must be strictly less than one. Similar arguments may
be used for the continuous-time case
componentwise and
if À1 <
0 then q
[2].
: e'
Consequently,
if Ài >
0 then
g(
e
The process is then called sub-critical,
critical or super-critical depending on whether
À1 is less
than, equal to, or greater
Lhan zer o respectively.
In the discrete-time super-critical case, Harris [9] has shown that if the process
is positive regular and non-singular, then
lim /,.(qn)
:
g,
(2.4.24)
r¿+oo
where qo is any starting vector
l"G):
"f("f"_r(")). This
in the unit cube of appropriate dimension
and
provides an algorithm for solving for the probability
of eventual extinction of the process. In Chapter 8 we derive an algorithm that
utilizes a similar equation to equation Q.a.2\ and then develop another algorithm
which converges to g in a significantly more efficient manner.
Chapter
3
Models of Macroevolution
3.1
Introduction
As already stated in Chapter 1, one of the fundamental problems facing evolutionary
biologists is to explain the diversity of life found on earth [22]. Attempts at providing
solutions to this problem should in principle provide some level of understanding of
the factors that have influenced diversification during evolutionary history. The
macroevolutionary manifestation of these factors results in changes in the rates of
speciation and extinction of species .77,22]. The consequence of this, as Mooers and
Heard [22] stated in their review article, is that "most biological taxa have arisen by
a branching process of descent with modification". In the context of developing a
suitable model to attempt to describe the macroevolutionary process, this statement
implies that a multi-type branching process provides a useful starting point, a point
to which we shall return.
3.2
Phylogenetic Trees: Species Relationships
The relationships between species in evolutionary history are represented pictori-
ally by a phylogenetic tree. Phylogenetic trees are constructed using information
that is obtained from observational data. This data, may be genetic or paleaonto22
CHAPTER
3.
MODELS OF MACROEVOLUTION
23
logical for instance. Since this data is incomplete, phyiogenetic trees can only be
inferred. These inferred trees may or may not represent the underlying actual tree,
the structure of which cannot be known [22].
A phylogenetic tree consists of a root node, internal nodes, Ieaf nodes, internal
branches which connect two internal nodes, or which connect the root node
to
an
internal node, and leaf branches that connect one internal and one leaf node, or
connect the root node to the leaf node if the tree has only one branch. The iengths
of the branches in well constructed phylogenetic trees should in principle represent
the age of the species. Figure 3.2.1 depicts a hypothetical phylogenetic tree for
extant species
A,B,C,D,E,F,G
and extinct species a,b and
3.2.1 one could conclude that species
other than say
A
and
C. Furthermore,
A and B
are more closely related
one could infer that
related to each other than say ,t' and G since
f'
c. Based on Figure
to each
A and B should be less
and G diverged at a later time. In
practice, such inferences should be made with caution, because the true tree with all
extant and extinct species and correct branch lengths is not known. In fact, there
is seldom enough information to be able to accurately include the extinct species in
the analysis.
There are a number of statistical and practical problems associated with the
reconstruction of phylogenetic trees. For a review of these consult [22]. However, despite these possible problems, phylogenetic trees can provide insight into
the macroevolutionary process.
3.3
Tree Topology
The topology of a tree is defined
122]
to be the branching pattern of that tree when
the lengths of the branches and the labels of species at the leaves are ignored. Thus,
any tree can be drawn in a topological fashion
equal; this is depicted in Figure 3.3.1. Trees
if all the branch
A
and
lengths are made
B are topologically
identical,
the only difference is that the branches of tree A have varying lengths whereas the
CHAP']:ER
3.
MODELS OF MACROEVOLUTION
b
24
c
a
AB
C
DEFG
Figure 3.2.I: A hypothetical phylogenetic tree
branches of tree
B
have identical lengths. The branch lengths of a tree represented
topologically do not reflect the ages of the branches.
Tree A
(Branch lengths and topology)
Tree B
(Topology only)
Figure 3.3.1: Two representations of the same tree
The topology or shape of a tree conveys information regarding the positional
relationships between species. The topology can also provide information on the
CHAPTER
3.
MODELS OF MACROEVOLUTION
25
propensity for speciation or extinction. The driving forces behind speciation or
extinction are likely to be complex and varied, for example, some may be biogeographical others may be genetic lI2, 22]. The consequences of these macroevolu-
tionary driving forces is that the shape of the subtrees differ; as a result the tree
is imbalanced. Imbalance can be quantified and there are a number of measures of
imbalance, each one with its own interpretation [4, 75, 22). Nonetheless the more
imbalanced a phyiogenetic tree is, the more varied the rates of speciation and ex-
tinction in different parts of the tree 1221. Changing conditions at different physical
locations influence speciation; evolution may either "speed up", "slow down" or
cease altogether
in these locations.
To
,/'
Figure 3.3.2: A tree with varying speciation rates
For example, consider the topology of a phylogenetic tree where the left subtree
undergoes speciation much more rapidly than the right subtree. An example of such
a tree is given in Figure 3.3.2. We have labeled the left subtree To and the right
subtree T1. The subtree Tohas undergone many more speciation events than subtree
{,
illustrating that varying speciation and extinction rates can have dramatic effects
on the topology and hence the imbalance of a tree. The greater the variation in
the rates of speciation and extinction within different parts of the tree, the more
imbalanced the tree
will be. The topology of the tree thus conveys information
CHAPTER
3.
MODELS OF MACROEVOLUTION
26
about the historical macroevolutionary process.
There has been considerable interest in studying the imbalance generated by
different probability models of tree generation [1,4,8, 10, 11, 12,\5,,22,26,30]. A
model of macroevolution provides a probability measure on the space of tree shapes.
As a result some macroevolutionary models may predict more balanced topologies,
whereas others might predict less balanced topologies.
To conclude this section, we define the concept of topological isomorphism. Two
trees are topologically isomorphic if by a suitable interchange of the left and right
branches at each node those two trees can be made identical. The trees depicted
in Figure 3.3.3 are topologically isomorphic since the first tree can be transformed
into the second tree by interchanging the left and right branches at nodes 0, 7,2,
and
3,
4.
1
2
4
Figure 3.3.3: Two topologically isomorphic trees
Note that in this thesis any two topologically isomorphic trees are considered as
having di,sti,nct topologies. Whereas in l22l any two topologically isomorphic trees
are considered as having the same topology. Furthermore, there is a subtle difference
in the use of terminology, what we call a topology is called a tree in[22] and
[30].
CHAPTER
3.
MODELS OF MACROEVOLUTION
27
3.4 A Labelling System for Binary Trees
rrVe defrne
three types of nodes for a binary tree (or for a general tree)
o internal nodes, also called branch points,
o extinct leaf nodes, and
o unstable leaf nodes.
The reason why we have chosen to use these three nodes types is due to the fact that
a significant portion of this thesis is concerned with demonstrating that the models
of Pinelis [26], who utilized this particular choice of node types, is encompassed
by the macroevolutionary model that we propose in Chapter 5. As a result, it is
necessary to go through and define branch types and nodes types in more detail.
An internal branch is deflned to be a branch that has completed its evolution
and is not a leaf branch. An extinct branch is a leaf branch that has completed its
evolution. An unstable branch is defined to be a branch that has not completed its
evolution. Thus an unstable branch will either generate a new daughter and become
internal or
it will become extinct.
time
t
f,
t,
Figure 3.4.7: An example of the evolution of an unstable leaf node.
CHAPTER
3.
MODELS OF MACROEVOLUTION
28
Internal nodes, also known as branch points, have a fixed position in the tree
and do not change as the tree evolves and their name suggests they are neither leaf
nodes nor root nodes. An extinct leaf node is the node
that is at the end of an
extinct branch. The position of an extinct leaf node is fixed it does not change
as
the tree evolves. The best way to explain an unstable leaf node is to consider Figure
3.4.7. In this flgure we depict a single branch. This branch is depicted at three
different times, t1,t2 and ú3. Since the branch is evolving, the position of leaf node
ly' alters from ú1 to
ú2
and finally to ú3.
At time Í3 the branch becomes extinct
the position of the leaf node is finally fixed and
it
and
becomes an extinct leaf node.
Thus an unstable leaf node is not fixed whilst a branch continues to evolve but it
becomes fixed if the branch undergoes a branch point and becomes an internal node
or if the branch becomes extinct the node becomes an extinct leaf node.
Flom a topological viewpoint however, the single branches in Figure 3.4.1 are all
identical,
time
it
makes no difference that the unstable node, Iy', was not fixed up until
ú3.
Here and throughout, we encase the labels of nodes in square brackets to ensure
that each node can be recognized without ambiguity. We begin labelling from the
[0] node. This node is either the unstable leaf node of the root branch, the extinct
Ieaf node of a single branch tree, or the first internal node of a tree that has at least
two branches. We later give a label to the root node, which is the parent node to
[0]. Supposethat
[rþ]:10,'ir,...,i^], where'ir,...,i^e
{0, 1}, is anode of abinary
1 be defined as the depth of the node [T/]. The node that
tree. Let þÞl :
^*
connected to the left of [r/], called the daughter node, is labelled,
lrþ,01
:
10,
i.r, . . .,
is
i*,0f,
whereas the node that is connected to the right of [T/], called the parental subnode
of [ú], is labelled by
lrþ,Il
:
[0,
rt,
..
.,i^,7].
CHAPTER
3.
MODELS OF MACROEVOLUTTON
29
As a matter of correct terminology, the parent node of [,r/]
:
[0,
'it,.
..
,i,^] is always
that node that is at a depth of rn with label [0,2t,...,i,n tf and all nodes, ltþ,"p],
where e¿ is an 7 x k row vector of ones, are the parental sub-nodes of lrþ] provided
that these nodes exist. Figure 3.4.2 depicts
a
binary tree topology with all its nodes
labelled.
I,l
0,1,0
,/0,0,,
0,0,0
'---ul,l,1
1,0,r
0,0,1
0,1,1,0
0,1,0,0,1
1,1
0,1,0,0,0
0,0,1,0,1
Figure 3.4.2: An example of the labelling of a binary tree.
Once again, let lrþ] : [0,it,.
..,i^]
be any node except for the root node of
a
binary tree. The function a has the following action on [ú],
a(?þ)
:
[0,
rr,
. . .,'i,,,_t],
that is, a(þ) is the parent node of [r/]. We hence label the root node of a binary
tree by a(0). The function d, on the other hand, acts on any internal node, lrþ1, o,
the root node a(0) so that,
0(rþ)
:
lrþ,r1,
and
e(o(o))
Suppose
that
[T/]
:
[0,2r,
...,i-]
:
[0].
is a node of a binary tree. The branch segment
between the nodes [r/] and lrþ,i^+t) is represented by,
[rþ],lrþ, i^+tl),
CHAPTER
where
3.
MODELS OF MACROEVOLUTION
'i^¡t e {0,1}.
Consider the branch
o if the branch is extinct,
30
(["(ú)], ['rl]), then,
write (t"(ú)] ,lrþl)("), and also write
we
lrþ)@
to denote
(n)
to denote
the extinct leaf node of that branch,
o if the branch is internal,
we
write ( [" (ú)] , [ri] ) (o) , and
also
write
[ú]
the internal node of that branch, and
o if the branch
is unstable we write,
(["(ú)], [rr])(") and also write
lrþl@ to denote
the unstable leaf node of that branch.
If a branch type is unimportant
Let
T
If
[r/] is any internal node of a tree
T,lhen the tree of topology,TW),
based on node [T/] can be written
denote the topology of a binary tree.
of this topology,
as
we do not specify a superscript.
the ordered set,
Tt
where Tþþ,ol und
T¡,¡,,11
4
:
{ ( t'(ú)1, lrþDØ, Ttþ,¡t, Tt r,rt},
are the topologies of the daughter and parental subtrees whose
first internal branch points occur at nodes [T/,0] and fy', 1] respectively, and for
single branch topology that is extinct we have,
Twt
:
{
a
([o(ú)], [,r])(") ],
or for a single branch topology that is unstable we have,
Tt
We say lhat
þt
: {(['(ú)], t'll)(") ]
T*, is the parent tree of the daught"r,
Tl,,þ,o1,
and parental,
subtrees. Consequently, at a branch point, say node ['rl], the branch ([T/],
refered
T¡,¡,,r1
[/,0]) is
to as the daughter branch and the branch (lrþl,lrþ,1]) is refered to
as the
parental branch.
Having discussed internal, unstable and extinct nodes we wish to introduce one
more node and branch type, called a quasi-stable node and a quasi-stable branch [26].
A quasi-stable node can be thought of as being similar to an unstable node, except
CHAPTER
3.
MODELS OF MACROEVOLUTION
31
for one important difference, a quasi-stable node can never become an internal node
or an extinct leaf node, therefore once a quasi-stable node is formed, that portion of
the branch becomes a non-extinct leaf branch so the node is never frxed. We denote
a quasi-stable node bv lrþ]n and a quasi-stable leaf branch as ([a(/)] ,lrþ])0.
3.5 Macroevolutionary
Models
One of the aims of the biologist is to decipher the possible causes of rate variation
and to understand how each leaves its "footprint" on macroevolution. The develop-
ment of stochastic models of the macroevolutionary process may shed some light on
the manner in which such complex systems have evolved over time. Models that act
as good starting points must have the flexibility
to account for any of the myriad
possible tree shapes that have been inferred from biological and/or paleontological
evidence. However, due to the complexity of the macroevolutionary process a balance needs to be found between the need to provide a sound underlying biological
basis and the need to provide algorithmic tractability. To attempt, a pri,ori,, to in-
clude all the known macroevolutionary factors into one model would prove to be
intractable.
Branching processes and in particular multi-type branching processes have been
utilised in biological applications for some time [14]. Mooers and Heard l22l and
then Aldous [1] proposed the potential use of a cIMMTBP in a macroevoiutionary
context while Pinelis [26] developed a model called the multi-rate model (MR) which
was based on the cIMMTBP. One of the major drawbacks to using the ctMMTBP
in a modelling context follows from the fact that there
seems
to be an insufficient
number of numerical algorithms from which to calculate the useful measures of the
model [5]. In fact the major problem with using the MR approach of Pinelis lies in
the fact that there are no reasonable algorithmic approaches except for the simplest
of model types. Dorman, Sinsheimer and Lange [5] have identified this problem and
provided a step in the right direction. They considered a ctMMTBP with Poisso-
CHAPTER
3,
MODELS OF MACROEVOLUTION
32
nian immigration and numerically integrated the Kolmogorov backward differential
equations. They then applied a finite Fourier transform to obtain the marginal dis-
tributions for the generating functions of the probabilities of particle numbers and
immigrant particle numbers. As a result, they were able to calculate the mean and
variance of particle numbers, and determined numerically the probability of extinc-
tion at time ú. Dorman, Sinsheimer and Lange [5] found that in the supercritical
case) as
failed
time gets large, the algorithm for determining the probabiiity of extinction
[5].
The remainder of this chapter is devoted to discussing some general probability
concepts for tree topologies in Section 3.6, followed by a more in depth look at tree
topological properties in Section 3.7, and then an introduction to Colless's measure
of imbalance in Section 3.8. Section 3.9 reviews one of the simplest and most studied
branching models, the birth-and-death model. In Section 3.10 the proportional-todistinguishable arrangements model (PDA) is also reviewed, and we show that the
subcritical birth-and-death model generates the PDA model. Finally, in Section
3.11 the most complex model to date is discussed, the multi-rate model, which is a
continuous-time Markovian multi-type branching process [26].
3.6 Probability Measures
and Tree Topology
The state space in which most branching processes are studied is the non-negative
integers for one-dimensional branching processes, or the space of n-dimensional vec-
tors with non-negative integer components for n-dimensional branching
12, 91. However, since we
processes
wish to model the macroevolutionary process, such state
spaces are not the most useful or
insightful. As emphasized previously, the topology
of phylogenetic trees reveals much about the underlying macroevolutionary processes. Consequently, having a process on the space of particle numbers is not
nearly enough, we need to be able to keep track of the history (lineage) of all the
particles in a branching process
if
we wish to use
it
as a model of macroevolution.
CHAPTER
3.
MODELS OF MACROEVOLUTION
ùù
Knowledge of the history of the particles allows us to map the realization of the
process
to the space of tree topologies. Thus, instead of analyzing branching pro-
cesses on
the space of positive integers we shall analyze branching processes in their
more natural format: on the state space of tree topologies which we denote by
11.
The classical branching process framework can always be recovered by counting the
number of leaf branches of a topology.
To be more precise, in this alternative framework, the evolution (ageing) of
a
particle traces out a branch of the tree whose branch length is the age of the particie
since
birth. In
new particles.
any realization of the pÍocess, a particle may die or give rise to
If this particle gives birth to new particles we consider this parental
particle as still remaining alive. As a result, each realization of the branching process
for all times generates a tree whose branch lengths are dependent on the ages of all
the particles. However, we are not interested in all this information, but rather
the topology of the tree as it evolves. We recover the topology of the tree by
applying a mapping from the space of trees to the space of tree topologies. Denote
this mappingby
M.
This mapping is a many to one mapping since there are an
uncountably infinite number of trees that all have the same topology but differ only
in their branch lengths. Note that as time evolves the space of topologies, lf,
is
exactly the same. What changes is the set of trees that map to each topology in lf.
3.6.1 Branch
Types and Associated Probability Measures
There are three important generic branch types that play an important role in what
follows. These three generic branch types are
1. extinct branches
2. unstable branches, and
3. quasi-stable branches.
CHAPTER
3.
MODELS OF MACROEVOLUTION
34
Let us expiain each one in turn. An extinct branch is generated when a particle dies,
that is, it
ceases
to evolve. An unstable branch is generated whilst a particle is still
actively evolving and capable of either giving birth to new particles or becoming
extinct. A quasi-stable branch is generated when a particle is not extinct but
is
unable to give birth to new particles. The use of the terms unstable and quas'i-stable
is borrowed from Pinelis [26]. This concept of a quasi-stable branch type (particle
type) is crucial to the modeling in Pinelis's paper [26] on the Multi-rate model which
we discuss later. From here and throughout we shall refer to branches and particles
interchangeably.
We can define different forms of the mapping "Al depending on the branch types
that interest us. Recall, that the mapping "ll4 disregards branch length; this is still
true of the variant mappings that we discuss here. The mapping that gives us the
topology of a complete tree for any ú is denoted by
M" : M.
The mapping that
gives the topology of the extinct portion of a tree for any ú by pruning all branches
except for extinct ones is denoted by
M.
The mapping that gives us the topology
of the unstable portion of the tree at any time
unstable ones is denoted by
M.
ú
by pruning all branches except for
The mapping that gives us the topology of the
quasi-stable portion of a tree at any time t by pruning all branches except for quasistable ones is denoted by
Mn. It is important to note that all these mappings map to
the same space 'lf independent of branch type. However, so that no ambiguity arises
when discussing certain models or the trees that are generated by those models,
o if T is the topology of the entire tree, we write 7",
o ff T is the topology of the extinct portion of a tree, we write 7",
o tf T is the topology of the unstable portion of a tree, we write T", and
o ll T is the topology of the quasi-stable portion of a tree, we write 7q
The branching process models that we analyze are continuous time models and so
the topology to which a tree is mapped will change in time. To illustrate this point,
CHAPTER
3.
MODELS OF MACROEVOLUTION
35
Tme
v
z
Figure 3.6.1: Same number of branches at different times does not mean topology
is the same
consider the tree depicted in Figure 3.6.1
.
This tree has evolved until Time
t:
z.
This tree consists of unstable and extinct branches but no quasi-stable branches.
Suppose we wanted to know the topology of the tree aI
t : r. At t : r
there are
four unstable branches and no extinct branches. The topology of this tree is shown
in Figure 3.6.2. Lt t
:
E there are also four unstable branches, but between
t : tr
and ú : y four branches have become extinct. The topology of the unstable portion
of the tree at
t:
A is also depicted
unstable portion of the tree at
portion of the tree at
t
:
in Figure 3.6.2. Note that the topology of the
U is
not the same as the topology of the unstable
t : t, not because of differing branch lengths, but because
the actual shape of the tree at those two times is different.
\Me shall
finish this section with a short discussion of probability measures. As
stated above, we think of the process as being a mapping from the space of rcaliza-
CHAPTER
3.
MODELS OF MACROEVOLUTIO¡\r
36
Tree at time x
Tree at time y
Figure 3.6.2: Same number of branches at different times does not mean topology
is the same
tions, or tree histories at each ¿ > 0, to the space of fixed topologies and that for
each ú
)
0 there are uncountably
Hence, the probability
infinitely many trees that map to a single topology.
that a realization of the process has topolo gy T at time
ú is
given by the measure of the space of all trees of differing branch lengths that are
T at time ú. In the remainder of the thesis we denote this probability by
p(T,t): p(T",/) and we say that this is the probability that a tree has topology
mapped to
T at time ú. The probability that the unstable portion
of a tree has topology
T
at
time ú is denoted by p(T",ú), and similar definitions apply for p(T",ú) and p(Tn,t).
In other words,
because there are four different mappings from the space of tree
histories of the process to the space of tree topologies there are also four different
probability measures for the process. These measures are determined by what por-
tion of the tree interests us, whether it be the entire tree, the extinct portion, the
unstable portion or the quasi-stable portion. Thus the mapping from the space of
tree histories to the space of topologies is determined by the measure that is being
used. For example, we are interested in the measure for the unstable portion of
the tree and its associated mapping,
topology
T at time t is p(7",ú)
Jv|,
then the probability that a tree will have
and this is clearly not necessarily the same as the
CHAPTER
3.
t,-7
Jf
MODELS OF MACROEVOLUTION
probability that that same topology has under one of the different measures.
It
is
important to also note that although the mapping removes all other branch types
except for the ones of interest in calculating the actual measure itself, knowledge of
the entire tree history remains important, and this will be evident when we discuss
the multi-rate model in later sections.
T € 'lf, is given by the number of leaf branches and is
denoted AV lfl. So if lfl: s we say thatT is of size s. Similarly,
The size of a topology,
.
lT" I is the size of the topology of the entire tree,
.
lT" I is tfre size of the topology of the extinct portion of a tree,
. lT"l is tne size of the topology of the unstable portion of a tree, and
.
quasi-stable portion of a tree.
lTo I is ttre size of the topology of the
Denote the subset of
lf
such that all
7
in this subset have l7l
:
s by '1f". As
before, this space remains invariant to the mapping used, so that exactly the same
topologies are in
lf" regardless of whether we are interested in T", 7", T" or Tq.
For the remainder of this chapter Iet z
trees that at time ú are mapped by
€ {a,e,u,q}. Let S!,, be the space of
M" to 1f". The probability that a tree is in 'lf"
at time ú is just given by the measure of the set S!,r, that is, the measure of the
set of trees that are mapped,by
M" to 1f", which is denoted tv p(lT"l : t,ú) for
t) : p(lf"l : s). We will also denote
simplicity. Furthermore, lim¿-oo p(lf " | :
",
this probability by p(11ã,ú), where we piace the superscript z on'1f" to denote the
fact that we calculate this under the measure that generates trees of type z.
At this point it is worthy of note that most of the analysis that is to be performed
in this thesis is not a transient analysis, but instead the analysis is performed
ú
---+
oo. In this regime, if the branching
process model
that
as
generates the trees
is subcritical, then the tree consists, almost surely, of a finite number of extinct
branches, or of a finite number of extinct and quasi-stable branches. The branch
types that are present as ú --+ oo should be ciear from the context.
CHAPTER
3.7
3.
MODELS OF MACROEVOLUTION
38
Some Topological Concepts
Denote the set of trees that are topologically isomorphic
J eßg)
then ß.(.7)
to T e T by F.(7). If
: F(7). If the daughter and parent subtrees at node lrþl are
not from the same topologically isomorphic class, we say that node
[T/] is an uneven
node. Let the number of uneven nodes in a tree of topology T be denoted by
It
,r.
is clear lhal e7 can be calculated 126l by
er
where
I{A}
:
I{F(T¡o,o) lß(T¡o,tl)}
*
.rio,o,
is the indicator function of the condition
If a tree of topology, T,
topologically isomorphic to
I ,",o,,,,
A, andrt
lf l:
(3.7.1)
1
then, €T
:
0.
has 67 un€v€n nodes, then there are 2'T trees that are
T,that
is, the set JF(Z) contains 2er ftees. Figure 3.7.1
depicts the two distinct topologically isomorphic classes of size
4. Class 1 consists
of one tree because there are no uneven nodes, whereas, there are four trees in the
second class because there are two uneven nodes.
Class
I
Class 2
Figure 3.7.7: The two topologically isomorphic classes of size four
The set of topologies of size s, 11", can be partitioned into its topologically
CHAPTER
3.
MODELS OF MACROEVOLUTION
39
isomorphic classes. These classes are disjoint, and span all of '1f". Thus, if we let
denote the z-th topologically isomorphic class, then'1f"
:
{Fr,",...,IFr","}, where ?l
is the number of topologically isomorphic classes, and is given, for s
T_t
'"
with 7r
:
I
g",r+ uL:t
D;:i
+
z\vt_
>
1, by
(J.7.2)
if s is odd,
1; see [30] and references therein.
The topologically isomorphic set,
logically isomorphic sets, ìFj,r and
rF,
_
" ¿'s
ToT"-¡) if s is even,
- +Ð;:iT¿T,-¿
I
IF¿,,
IF¿,,
can be constructed by combining two topo-
1F¿,"-¿,
Í t(["(0)], [01¡l';1, Fr,z, Fø,"-r] U {(["(0)], [01;tzl, F*,"-,, F¡,,] if F7,r lFn,"-¿,
: F*,"-,,
Fr,,"-r]
if F7,r
ìa
)t t F¡,r,
{(tr(o)1,
,.. \-/)tL [01;tzl,
JlLj K,s-L)
J,t
(3.7.3)
for some j e
{7,2,...,T¡}, andk e {1,2,...,7"-'t}.
This concept of topologically isomorphic class is of course a purely topological
concept. rñ/hen the actual probability measure along with its associated mapping
is important, we will label topologically isomorphic classes to reflect this. In other
words,
if the current
measure and mapping generates trees of branch type z, we
label the topologically isomorphic class,
3.8
IF¿,"
by
ß-f,".
Colless's Index of Imbalance
As we have stated previously, there has been considerable research performed on
gaining some understanding of the imbalance of phylogenetic trees through the process
of macroevolution. Measures of imbalance have therefore gained a prominent
piace in the study of macroevolution [1, 11, 15,22,26] and the degree of imbalance
of a topology can be quantifled using a variety of indices, see for example, [15]. The
one that has been most utilised is Colless's index of imbalance
a topology,
T of size lT l. Colless's index, I.(T),
for
T
I" 14,22]. Consider
is the total of the absolute
value of the difference between the number of leaves of the daughter and parent
CHAPTER
3.
MODELS OF MACROEVOLUTION
40
subtree at each and every node. Let the space of internal nodes of,
T,
be denoted
byIBz. If [?i] €187,then,asbefore,thedaughtersubtreeof lrþ]isdenotedbyTbp,ol
and the parental subtree of l',þl is denoted by Tþr,rl. Coiless's index of imbalance is
defined to be
r.(T): t
llrvp,,tl
,þeßr
I
I
I
I
I
I
I
I
I
I
t0,ll
I
I
I
I
I
I
T
.tt 1l
I
I
I
I
I
I
T
(3.s.1)
I
/
I
- l7rr,,lll
T
*iIo'0, 1l
I
ï
[0,0,1
T
I
I
I
t_
o;1 1l
I
I
I
I
[0,1,0]
I
T
ï
T t0,01
ïo,tt
Figure 3.8.1: Colless's index of imbalance for two trees
We now calculate
I"(T) for the two topologies depicted in Figure 3.8.1.
The
daughter and parent subtrees at each node are labelled for both trees. Colless'
index of imbalance for the first tree is,
I.(T)
D
,þeßr
llr*,,r1-
l7rø,,rll
- lzrr,rtl
2+l
lzro,ql
2
1
+
+
lzro,o,q ¡
1
1
- l7¡o,o,rr | *
:0,
I
|
lzr,.r
I
-
lzr.,,,,r
I
I
CHAPTER
3.
MODELS OF MACROEVOLUTION
and for the second tree
I"(T)
fu
47
is,
ÐllTr,,rl-
lzrø,,rll
,þeßr
lTppt
-
Tto,tll +
3-7 + 2-7
+
lzro,o,or ¡
1
1
-
lz¡o,o,rr I |
* lzrr.,, ¡ |
lz¡.,r,r,,r
I
I
-3
In fact, Colless' index of imbalance ranges from zero for the most balanced tree to
(lf l- 1)(l7l -2)12 for the most imbalanced
tree and so
it
can be normalised
if
one wishes to do so.
Colless's index of imbalance for trees can be calculated recursively from the
values of the two lower order tree shapes, namely the parent and daughter subtrees
at [0], thus,
I"(T)
:
I"(T¡o,o) + I.(T¡0,4)
r ls - 2ll,
(3.s.2)
where I.(T¡o,o) and I.(T¡s,r1) ut" Colless's indices of imbalance for the daughter and
parental subtrees at [0], with l7¡¡,0¡l
:
:
p(T",t : lT' | :
")
topology T" at time ú conditioned
on
I and lTfr,rll : s-l leaf branches respectively.
p,(T',t) be the probability that a random tree has a
Let
p,(T")
In addition, we denote the
ElI",úls],
and deflne
it
:
having size s. We write,
]im
p"(7",t).
expected value of 1" for trees of size s by,
Eill.,t] :
it by,
E!ll.,tl: Ð t"Ø")p"(7",t).
Tz€T,
(3.s.3)
The value of EilI",ú] is dependent on the stochastic model of macroevolution. Since
we are mainly interested in the behaviour of the models as ú --+ oo let,
EílI
"l
:,li1 t\ilI., tl,
and so the expected value of Colless's index of imbalance becomes
EXlr.l:
t"(r")p"(T")
Ð
T.€Ts
(3.8.4)
CHAPTER
3.
MODELS OF MACROEVOLUTION
FYom a computational
42
point of view, these expressions require a summation over all
the possible topologies of a given size. We can simplify this expression by noting
lhat I.(7") is the same for all trees within a topologically isomorphic class. This
is not difficult to see, when one considers that topologically isomorphic classes are
related by rotations at uneven nodes, these rotations only interchange the daughter
and parent subtrees and so do not affect
I"(T").
Therefore the mean imbalance can
be re-written in terms of topologically isomorphic classes. Let 1"(ß.¿,") denote the
imbalance of each of the trees in
IF¿,",
then we can re-write the mean imbalance
Ts
E"lr"l:
Ð
r.(Fi,)p,(Fi,),
where, p"(Fí,") is the conditional probability as
f
---+
as,
(3.8.5)
oo that a tree is mapped to
has size lT" | : t. rü/ritten in this way, the sum is not over
all topologies but over all topologically isomorphic classes of a given size and so is
T' e IFl," Biven it
computationallv more efficient.
3.9 Birth and Death Model
One of the simplest models of macroevolution is the birth-and-death model (BD)
This model has been studied extensively in the probability literature in a wide range
of contexts. In the context of macroevolution,
it
is a common assumption
lI,
22, 26,
30] that the rates of speciation and extinction for all the branches are the same. In
this case the model is called the equal rates Markov model [1, 8, 10, 26, 30]. However,
we shall always refer to
it
as the constant-rates birth-and-death model
(crBD) since
we believe this to be a better description of the process. Under this assumption the
qualitative evolution is such that any branch of the tree has probability Àdt of giving
birth to a daughter branch in a time interval dt and probability
¡L"dt
of becoming
extinct in a time interval dú, independently of the rest of the tree.
The crBD model has only two types of branch states, a branch is either unstable
or extinct. We partially follow the analysis of this model as is given in [26, Appendix
CHAPTER
3.
MODELS OF MACROEVOLUTION
43
A]. In [26, Appendix A] the topology of a tree is given by only the unstable portion
of that tree; all the tree's extinct branches are therefore pruned. Lef
p9Ø) be the
probability that a time ú there are no unstable branches. Note the use of the empty
set symbol to denote that such a tree has no topology since there are no unstable
branches. The probability that a tree will be extinct by time ú is given by,
: |¡t ø"*v( - tr + ùr)dr + |¡L lexp ( - tr + p,)r)e!,t - r)p(Ø,t - n)d,r.
Jo
Jo
p(Ø,t)
(3.e.1)
This equation can easily be understood by noticing that a tree will become extinct
if
the parental branch, ([r(0)], [0]) becomes extinct within (0, ú] before undergoing any
births, or that there is a birth at node [0] at time
r
subtrees both subsequently become extinct by time
and the daughter and parental
ú.
Now the probability that no events occur by time ú, and so the tree consists of
only a single unstable branch ([r(O)], [01;t"1, is given by, exp (-
tf + ùt).
there is an extra term which is due to the application of the mapping
if
However,
M,
so tItaL
any branch point occurs where one branch becomes extinct by time ú then
it
is
pruned and therefore the tree retains a one unstable branch topology. Therefore the
correct probability that a tree will have a topology consisting of only one unstable
branch by time
p(lr"
|
:
1, ú)
:
ú
is given by,
"*p
(-(,1+p)t)+
À
exp
(- (À+ ø)r)zp(lr"l
:
1,
t-r)p(Ø,
t-r)dr
More generally, the probability that a tree commencing from [r(O)] will be mapped
to a topology T" that has
p(T',t)
:
lf"l
> 2 at time ú is
fo' ^"*o(
+p(Trt,o¡,t
- (À + tùr)(ro("",t - r)p(Ø,t - r)
-
r)p(Trt,t1,t
- ,))ar,
(3.e.2)
where Trt,01 and Trt,r, are the topologies of the daughter and parental subtrees com-
mencing from node [0]. Equation (3.9.2) has the following interpretation: at time
0 the tree begins at, [a(0)], and in the interval
(r,rldr)
the root branch under-
CHAPTER
goes a
3.
MODELS OF MACROEVOLUTION
birth with probability ¡"-(\+ø)r¿r.
44
FYom this branch point there are two
possibilities:
f
.
in the time
intervalt-r
one of the daughter or the parental branch can become
extinct while the other evolves into a tree with topologv Tu, or
2. in the time
intervalt-r
that has topology
the daughter branch eventually evolves into a subtree
Tfr,o1 and
the parental branch evolves into a subtree that
has topoloSV Trt,t1. The daughter and parental subtrees evolve independently.
Due to the independence of the evolution of the parent and daughter branches in
both cases, the probability of each of these two scenarios is just the product of
the probability of each of the individual topologies, that is, 2p(T",t
and p(Tfr,sr,t
-
r)p(Trt,t1,t
- n).
\Me then integrate
r)p(Ø,t
-
- ")
z from 0 to ú as the original
branch point can occur at any time in that interval. Thus the probability of a tree
having topology
T" at time ú can be determined recursively from the lower order
tree probabilities using equation (3.9.2).
Harding [8] studied the case of the pure birth process (where
p:
0) by consid-
ering the embedded process only at birth points. He showed the probability that a
tree of size s beiongs to the topologically isomorphic class F(T") is given by
e(F(r,)): 4l$Ï-{e(n'(z¡ö,,q))e(m1z¡0,,,1)),
(3.e.3)
where F(7';,01) and tr(7¡fi11) ur" the corresponding topologically isomorphic classes
for the daughter and parent subtrees at node [0], respectively.
Pinelis [26, Appendix A] tried to prove that this relation still holds in the transient crBD model. To show that equation (3.9.3) was valid in this regime, he first
wished
to show by induction that a random tree, commencing with one branch,
evolves into a topology
T" of size lT"l :
"
by time t, with probability
p(7",t): n(T")p(Ti,t)
(3.e.4)
3.
CHAPTER
MODELS OF MACROEVOLUTIO¡\I
Pinelis made the assumption that the factor
o(7,)
with n(T")
:
1
if lf"l :0
-
*(T")
45
obeyed the following equation,
o(Tö,0)"(Tö,').
lr"l-1
(3.e.b)
'
or 1. Pinelis then assumed that equation (3.9.4) is valid
for all topologies with sizes less than s. To show that equation (3.9.4) was valid for
T"
of.
size s, Pinelis substituted
p(T",ú) from equation (3.9.4) into equation (3.9.2)
to obtain,
n(T")p(T!,t)
tr + p,)r)(ro(r",t
^"*o(+p(Tö,ot,t - r)p(Trt,r1, ú - r)) at
fo'
(
r)p(Ø,t
- r)
- tr + tt)r) (zoçr"¡eçTy,t - r)p(Ø,t - *)
/'
^.*o
+
(Tfi,01)n (Ti, t
n
-
- r) n (Trt¡1 )r (11Í- *, t - r)) dr,
(3.e.6)
where the daughter, Trt,o1, and parent, Tö,l,subtrees at node [0] are of sizes k and
s
- k respectively. The flaw in going from equation (3.9.2) to equation
that in equation (3.9.6) Pinelis
(3.9.6) is
used
p(T", t)
:
n(T")p(Ti, t),
(3.e.7)
on the right hand side, which was exactly what he was trying to prove. He assumed
it to be true on the right hand
side in order to eventually show that
it
is true on the
left hand side. It is our purpose to show rigorously that equation (3.9.4) is indeed
the solution to equation (3.9.2).
Theorem
then
t
If, p(7",t), giuenby equati,on (3.9./)
i,s a
solutzonto equati,on (3.9.2)
n(T") must sati,sfy,
o(7,) -
wi,th rc(T")
:1 if lT"l: O,t.
nØö,0ìnØö,tt).
lT"l-r '
(3 e.8)
CHAPTER
Proof
:
3,
MODELS OF MACROEVOLUTION
We begin the proof by assuming that
46
p(T",n), given in equation (3.9.4)
is a solution to equation (3.9.2) and then showing that n(T") must satisfy equation
(3.9.8). Consequently, substituting equation (3.9.4) into equation (3.9.2) gives,
n(T")p(T!,t)
:
lo'
(r + tr)r) (z*çr"¡eçri,t - ,)p(Ø,t - ,)
^"*r(
+
(Tfr,o) n (Ti, t
n
-
r)
n (Trt,t1
)r
(113-
r, t
- r)) dr
.
(3.e.e)
At this point
we cannot simplify the right hand side any further.
As has been stated earlier, the probability that a random tree will have size s
at time
ú is
p(11f,ú). Hence by the same argument as that used to derive equation
(3.9.2), we have
p(Ti,t)
one can
lo'
^"*r(
s-1
- tr + t)r) (rrq,t - n)p(r'",t - ,)
+ t p(Ti,t - r)p(T",-,,t - r))dr.
(3.e.10)
/
t:I
find the exact expressions forp(T'|, t) in[26, Appendix A], where the original
derivation of these equations was given in [13]. Using these exact expressions in [26,
Appendix A], one can easily deduce that the product of p(Tf ,t-r)p(T""_t,t-r) is the
same for all I
e
{7,2...,s- 1}. Consequently, we choose some k e {I,2...,s-
1}
and equation (3.9.10) becomes
p(Ti,t)
:
+ tòr) (rre,t - r)p(r!,t
lo' ^"*o( - fr
+(, - 1)p(Tit,-r)p(Tï-t,t - r))ar.
Multiplying the above equation by n(T")
n(T")p(T!,t)
:
o(7")
+
(s
-
À
(À
+
¡t)r)
( 2p(0,t-")p(Ti,t-")
- n)p(T""-*,t - r))dr.
If we subtract the above equation from both
fo' ^"*o
(3.e.11)
gives,
exp(-
1)p(lli,t
- r)
sides of equation (3.9.9) we obtain,
(_-Q+¡lr)e$i,t-r)p(T!-n,t- r)(n(T¡o,op)n(T¡o,qu)-rc(T')(s-r))a, :
(3.e.12)
o
CHAPTER
3.
MODELS OF MACROEVOLUTION
47
Since the exponential function is always greater thanzero,
andp(Tft,t-r)p(T""_¡,,t-
ø) is non-zero, (otherwise only extinct trees would be possible), we have that
n(Trt,sì n(Trt,t1)
- n(7")(s - 1) :
(3.e 13)
o.
Re-arranging the above equation then gives us equation (3.9.8). Furthermore, n(T")
is well deflned, since
Tu
:
is the unique representation of
{(["(o)],
[01;tzl,
Tfi,s1,Trt,rÌ,
(3.e.14)
7".
Finally, equation (3.9.2) is the integral form of the Kolmogorov backward equa-
tion, and therefore has a unique solution, so the solution to equation (3.9.2) given
by equation (3.9.4) subject to equation (3.9.8) is the unique
Remark L
Suppose
thatTi andTf
solution.
r
are'in the same topologi,cally i,somorphtc
class.
Interchangi,ng the daughter subtree wi,th the parental subtree and u'ice l)ersa at an
uneuen node, say tþ, does not affect the product n(Tfi,s)rc(Tfi,r1). Thus
"9,|)
and therefore p(Tf
,t) : p(Tf ,t).
o(Tf):
So all the trees i,n a topologi,cally i,somorphi,c
class are equi,probable.
Corollary 2 The probabi,li,ty of obtai.n'ing a topology from the topologi,cally 'i,somorphi,c classFf,', condi,ti,onal on the si,ze of the tree bei,ng s, at ti.met, i,s gi,uen by
p"(lFi", t)
where
:21{ri't¡uy'"-'¡Pt(Fi't't)!=¿!Fi'"_.t't),
(3.9'15)
p,(.,,t) represents the condi,ti,onal probabi,li,ty wi,th respect to tree si,ze r and
I
represents the si,ze of the left-hand subtree.
Proof :
Let
T" e F..1,". The Remark immediately
following the proof of The-
orem 1 tells us that the probability of each topology
i € {7,2,...,7"}.
in IFi, is uniform, for all
Therefore, as there ate 2'ru topologies in 1F1", the probability
that a ranrlom tree is generated with a topology from IFl" at time ú is given
p
(Fi,",
t)
:
2'r" n(T")
p
(T!, t)
by,
(3.e.16)
CHAPTER
3.
Now let rc(F'i")
MODELS OF MACROEVOLUTIO¡ü
:2'ru o(T"), so equation
p(Fi,,,t)
:
(3.9.16) becomes,
n(Fi,,)p(11i,
If we condition on trees of size s at time
48
ú, we
(3.e.17)
¿)
obtain,
p(Fy,,,t)
P"(Fi,,,t)
p(Ty,t)
rc(F'i").
Suppose that Trt,01
e Fi,t and Tfr,t1 €
(3.e.18)
Fl,"_¿, and that Tö,oy Tö,rl have e¡ and
e1
uneven branch points respectively. Then, IFi¿ has 2'o topologies and FÍ,"-¿ has 2"
topologies and so,
"(FT¿)
:2',o
n(Trt,o),
(3.9.19)
and
o(F"*,,
-) :
2" n(Tö¡l)
(3.e.20)
.
Substituting the above into equation (3.9.8) gives
K(F'i")
2'ru
_ t
n(Fi,)n(Fi,"_,)
2ro*q
s
-
(3.e.21)
1
However, recall that,
€ru
:
I{Fi,t +Fi," ,} *.0 +.r,
(3.e.22)
and so after some re-arrangement, equation (3.9.21) becomes,
rc(F'i")
: 2I {Fi,¿*Fr,"-,t "(trT'] 1(Tä"-' )
(3.e.23)
Substituting equation (3.9.18) into the above equation we finally obtain
p" (F-i",
t)
:
2I
{Fi,t¡ry," -,¡ n
(Fi't'
t)
!
=t
!Fi' - n t),
(3.e.24)
and the corollary is proven.
Remark 2 Notethat equat'ion (3.9.18) tells us thatp,(',t)
or t.
zs actuallE i,ndependent
CHAPTER
3.
MODELS OF MACROEVOLUTION
49
Example L Calculati,on of probabi,li,ti,es condi,t'ioned on trees of si,ze I
To calculate p"(lF.i") we use equation (3.9.23) directly and then use equation (3.9.13)
to identify p"(F'i"). Denote the
space of class 1 topologies (see Figure 3.7.1) by Fi,¿.
The daughter and parental subtrees of this class are identical and we denote them
by lFi,z. Since the two subtrees come from the same topologically isomorphic class,
we obtain from equation (3.9.23),
rc(F.ia)
_ o(Fi,r)r.
The topologically isomorphic class IFf,, has representation,
ETp
:
{(['(o)],
[01;r';1,
IFir, ìF'ï,r],
sot
"(Fir)
:rc(lFi,r)2:1
Hence,
n(Fi,t)
:713,
and so by equation (3.9.18),
pq,(Fi,+,r)
:
å
The space of class 2 topologies of size four is denoted by Fi,+, and has represen-
tation,
Fï,+: {{(["(o)], [01;rrr, tri,r, Fï,,] [J{tt*tolt,
Since we calculale
n(Fi)
[01;t,;r,
Fï,,,
Fï,r]]
recursively, we need to calculate Fi,s first. The space ìFf,,
has representation
Fï,s
:
{{(t'(o)1,
[01;rtr, trï,r,
Fï,,] U{it"(o)1,
SO
"(Fi')
2
rc(Fir)rc(F'i,)
2
1
2x r:7'
[0])(o),
ffi,,,
Fï,r]]
CHAPTER
3. MODELS
OF MACROEVOLUTION
where in the second step we have used
"(Fi,r)
50
: K(Fià:
9.2---
1, as above. Therefore,
t2
tÐ
ÙJ
Thus,
Pa(Fi'n't):?
.\+) t
3
Clearly,
r+(Fi,n,t) + p4(Fi,s,t)
:
1
I3
+
2
f3
:
I,
since there are only two classes of size four.
In Section 3.10 we discuss another simple macroevolutionary model, the proportionalto-distinguishable arrangements (PDA) model, which gives us quite different prob-
ability distributions.
3.10 Proportional-to-Distinguishable-Arrangements
Model
The proportional-to-distinguishable arrangements (PDA) model is defined such that
each distinguishable arrangement (DA) of the species of a tree of size s is equally
probable. A distinguishable arrangement of s species is an assignment of labels on
the s leaf branches that is not equivalent to any other arrangement. Two arrangements are non-distinguishable if by a suitabie permutation of the uneven nodes, the
labels and topologies can be made identical. Figure 3.10.1 depicts four trees, trees
1,2
and 4 are DAs, whereas trees 1 and 3 are not DAs. Under the PDA model each
of these three distinguishable arrangements are equally likely.
The number of DAs of a given set of s species that generate a phylogenetic
tree that belongs to a particular topologically isomorphic class has a direct correspondence to the number of topologies within
that class. It can be shown that the
number of ways of relabeling a tree of topology
T of size s is given
by,126, Appendix
CHAPTER
3.
MODELS OF MACROEVOLUTION
AB C
D
C AB
D
51
AC D
A
B
CD
B
Figure 3.10.1: A distinguishable arrangements example
Al,
o(T):
#r*,
where e7 is the number of uneven branch points of the topology
(3.10.1)
7.
This is just
the total number of tree labelings, s!, divided by 2 to the power of the number of
branch points that are not uneven ,2þ-1)-et , since swapping labels between subtrees
at those branch points do not yield distinct labelings. Interestingly, l/u.(z)
:2'r
and
so considering any other topology in ß'(7) does not add any more distinguishable
arrangements and hence one can write,
ø(m'(7))
: a(T).
(3.10.2)
The above equation demonstrates the relationship between the number of DAs of
a particular topology and the number of trees within that topologically isomorphic
class.
Example
2
The number of DAs
for
trees of s'ize four
CHAPTER
3.
MODELS OF MACROEVOLUTION
52
There are two topologically isomorphic classes of trees of size 4, see Figure 3.7.7.
The trees from class
this class are given
t
have no uneven branch points, thus the number of DAs of
by,
¿(F'r,¿)
AI
:
,r:3'
Trees from class 2 have two uneven branch points, hence the number of DAs for
class 2 is given
bv
ilrz
:
:72.
'23::
a(Fz,ò
Thus the total number of DAs for trees of size 4 are 15.
We can use the above example to show how the probability distributions) con-
ditioned on tree size differ between the crBD model and the PDA model. Now,
as
already stated above, each DA of a given size has an equal probability of occurring,
so using the example above, we find that,
31
155
724
P+(Fz,+) :
155
P+(Ft,+)
:
(3.10.3)
(3.10.4)
since there are 15 DAs for size four Comparing these values to those of the crBD
model given in Section 3.9, where
P¿(Fi,s,t)
:
1
(3.10.5)
,
J
2
n+(Fi,n,t)
(3.10.6)
o
L)
we find that the PDA model tends to allocate higher probabilities to topologically
isomorphic classes with higher imbalances, because these classes
in
general have
more uneven branch points and therefore more topologies than the more balanced
classes. We see from this example that, under the PDA model, the probability of
each topology of size four is the same, namely 1/5.
The number of topologies of size s,
l/", is given by,
s-1
¡/":t&ÄL
i:7
¿,
(3.10.7)
CHAPTER
with
3.
MODELS OF MACROEVOLUTION
l/t : 1. It is easy to prove this using induction.
53
The first few terms of the
series are,
l/rlû :
N2
1
¡ñ^h I N2N1:2
^I3
N4
¡ú^h +
¡/5
¡i1¡/4 + ¡/r^¡3 +
N2N2
+
1V3¡f1
^IB¡/,
Theorem
3
:2*|
l2 :5 and
+ ¡/4¡ú : 5 J- 2 + 2 + 5 :
14
oo
r/(z)
:I*,":å(t
_
n:I
1fit-nÐ),
(3.10.8)
for0<r<714.
Proof :
To prove equation (3.10.8) we first muitiply equation (3.10.7) by
then sum from 2 to infinity and by noting the boundary condition
l[ :
1
r"
to obtain,
oo s-1
¡/(") : "+Ð!lr,rrL-r""
s:2 i:7
oo s-1
"
+D\N,roN"-rr"-i
(3.10.e)
s:2 i:L
The order of summation in equation (3.10.9) can be swapped to obtain,
l/(z)
:
ôo
oo
N,-rr"-i
Ð
"+DN,ro
i,:7
s:i*1
(3.10.10)
One can perform a change of variables in the second summation of equation (3.10.10)
with the result that
l/(r) :
"+DNu"nDN,!x':l/(')
i:l
s:1
: r+N2(r)'
(3.10.1 1)
One can solve equation (3.10.11) using the quadratic formula to obtain,
(3.10.12)
Of these two solutions we choose
(3.10.13)
CHAPTER
3.
MODELS OF MACROEVOLUTION
since \Me do not want ,A/(r)
>
54
1
Corollary 4
n, (2s - 1)!!^"
'
,n":
> 1.
- -l! r2"-',s
Proof :
(3.10.14)
Expanding equation (3.10.3) using a Taylor Series about
applying induction yields equation
r:
0 and
(3.10.14).
I
More generally then, if we condition on a tree size of s, the probability, under
the PDA model, of any one particular topology is 1/^L, independent of the actual
topology. Thus, if there are more topologies in a particular topologically isomorphic
class, the higher the probability
topology from that class. Let
that a random tree will be generated that has
ly'B,o,,
denote the number of topologies in
p"(lF¿,")
and so the more the topologies in
trees in
IF¿,s
ly'B'n,"
:
IF¿,",
a
then,
Np'"/N,,
the higherp"(lF¿,"). Note that inl22] all the
are considered to be the same topology. Consqeuently,
in
[22] p"(F'¿,") is
the probability that a random tree has the topology represented by the trees in
1F¿,".
Defined in this manner, the trees in the PDA model do not grow under some
stochastic dynamics. Attempts have been made at giving the PDA model an "evo-
lutionary" explanation; evolutionary in the sense of the temporal evolution of
a
stochastic process 126,,341. In this section we shall give an alternative model that is
simpler than that in [26], with the details given in126, Appendix A].
To do so, we would like to understand the connection, if any, between the the
crBD model and the PDA model. At first glance
it appears that
they have no rela-
tionship to each other, however this is not the case. The transient crBD model of
Section 3.9 gave us expressions for the probability that a random tree was generated
with a specific topoiogy at time ú. These topologies consisted entirely of unstable
branches; all extinct branches were pruned from the tree. If we consider the asymp-
totic version, that is, as ú ---+ oo, in the subcritical crBD case trees are generated
CHAPTER
3.
MODELS OF MACROEVOLUTION
with only extinct branches and so from this model
55
we obtain an entirely different set
of results for tree shape probabilities. Indeed, the limit of p"(T',ú) in the subcritical
crBD model as ú --+ oo will be shown to be identical to p"(T) from the PDA model,
for the same underlying topology 7.
The probability,
p(lf"l :
0), that a tree has no extinct branches as ú --+ oo
is clearly zero, since in the subcritical regime trees with no extinct branches exist
only on a set of measure zero. The probability that a tree has topology,
([r(o)],
is given by,
[01;r"1,
p(([o(o)],
lim
[01;r"r;
ú+oo
pexp
t;
]*T.
p
\+
The probability that at time
T" with lf"l>
p(T',t)
T" :
:
úa
(-
tr + p,)r)dr
(- + p)¿))
,,(r - ""n tr
(3.10.15)
tt
random tree will evolve into a tree with topology
2 is given by,
) exp ( -
This can be explained
as
tl ¡
¡lr)nØ¡F,01,t
(3.10.16)
follows. At time 0 the tree begins with a single branch which
then undergoes a birth within the interval (r, r
¡1,)r)dr. In the interval
- r)p(T¡6,t1,t - r)d,r.
(r,t]
I
dr) witln probability
À
exp ( -
(f +
the daughter and parental subtrees at node [0] evolve
into trees with topologies 7¡fi,01 and
T¡S,tl
respectively. Since they evolve indepen-
dently, the probability of this evolution is given by the product of the probabilities
of the daughter and the parental subtrees. Finally, because the original branch point
can occur anywhere in the interval (0, ú] we integrate ,r over that interval.
Theorem 5 For the subcri,ti,cal crBD model, a random tree euentually
topology,
euolues
to a
T" of s'ize s, wi,th probabi,l,itg,
p(T"):
tn the li,mit as ú ---+ oo.
,.
,,
(À ^'-t{
+ ¡t¡2"-t'
(3.10.17)
CHAPTER
3.
Proof :
We shall prove equation (3.10.17) using induction.
s
:
MODELS OF MACROEVOLUTION
It
is clearly true for
1 because from equation (3.10.15) we have
p(T"):
We shall now define
Suppose now
:{
(3.10.18)
( p(r",t)
ú > 0.
r\- '"'' if--":")
|. 0
(3.10.19)
if ú<0.
that equation (3.10.17) is true for all l1 s. Note that for a topology,
of size s * 1, the daughter subtree
subtree, Tl3,r1 t, of size s
ú --+
-+.
¡_r þ
p(T',ú) for all ú, such that,
p(7",¿)
T'
56
f
1
- l.
7¡fi,0,
is of size
I<I(
s and the parental
Now taking the limit of the above equation
as
oo we have,
p(7")
fú)
: ]iU
À e"p ( - tr + ¡')r)ø6¡3,01,t - r)p(T¡|,t1,t
- r)dr.
t_x, J/
(3.10.20)
0
The functions
p(X",t-r)
arc bounded on any compact set, and their
limit
as
f
---+
oo
exists. In fact,
:
J$p@",t - r)
]*p(x',t - r),
sinceú-r>0andso,
p(x")
lor lX"l
: Ì5-O(x',t - r) : }*p(x',t - r) :
rl-1.-l
ûfu,
: I < s, by the induction hypothesis. Therefore the Dominated
Convergence
Theorem implies that
p(7") : gT
/t-æ Jo
:
:
À
Ir* ^"*p
exp (
(
- ir ¡ ¡r)r)øe¡\,o.,t - r)p(T¡|,4,t - r)d,r
- (\+ ¡-t)r) ]5,¿p{ó,or,t - r)n(T¡1,4,t - r)d,r
(-
tr + ¡L')r)d'r
+ ¡r;zt"+1-¿)-1
l,* lexp (-
FØr3,ot)FTri,,',),{ )exp
¡t-t /
(À
+
¡s-t rs+r-t
¡1¡zt-t (À
Às-1ps+1
(À+ p)'"
I,* Àexp (-
t,t + p,)r)dr.
tl
+ p,)r)dr
(3.10.21)
3.
CHAPTER
MODELS OF MACROEVOLUTION
57
where in the fourth step we used the induction hypothesis as both trees are of size
at most s. Performing the integration yields
p(rir): (À,. +,^"t,,*1 ', ' ,
(
P¿;z{"+t;-r'
as
J.lo.22)
required.
Remark 3 The
r
aboue theorem shows that the probabi,li,tE of a random tree i,n the
subcri,ti,cal crBD process, euolui,ng to a topologu,
T' , of si,ze s * 1 zs d,ependent only
on its stze and not on i,ts topologg as t
Thi,s i,s i,n contrast to the trans'ient
---+
oo.
model, where the probabi,lity that a random tree euolues to a topology,
bll t'ime t,
n(7").
depends on
Theorem
T"
6
oo
ts-1.-s
(3.10.23)
Ðrr¡#='ÄL:1
Proof :
Noting that
)s-lps
À¡t \"
À \OltPi
eTtF"-':^+p(
)
and writing,
rù')' *"
(,^
å
À+ p
n(^ yæ)''') ß,0,n)
2^ (' - (' À+ p
Ë+(55;""
where equation (3.10.3) was used
À
in the third step. After
some simple algebra
equation (3.10.24) can be shown to be equal to one thus proving the
Corollary
7
The the li,mi,t as
t
---+
theorem. r
æ of the subcriti,cal crBD model condi,ti,oned
tree s'ize generates the PDA model as
t -+ æ.
on
CHAPTER
3.
Proof :
The probability, using the crBD model, that a random tree eventually
MODELS OF MACROEVOLUTION
evolves into a topology,
T" of size s is given
58
by,
't.s-1 ..s
p\t') oEF;rr.
t
A
-P\
l-L
l/".
Let the number of topologies of a given size s be
Now, since the probability of
any topology depends only on size in the subcritical crBD model as
I
--+ oo, we have
that
p(r:)
: *(^T;
)y:"_,
Finally, given that the tree is of size s, the probability of obtaining a particular
topology, T" , p"(T"), is
p"(T")
:
'lT)
p(r3)
À'-
1
s-
À
^t1
¡/"
The above expression is exactly the probability of a random tree having any topology,
conditioned on tree size, in the PDA
model.
r
The crBD interpretation of the PDA model considers extinct trees;
it
gives more
weighting to less balanced topologies, in comparison to the transient crBD model
that considers unstable branchesl. Topologies that are less balanced belong to topoIogically isomorphic classes that have a greater number of topologies, since there are
a higher number of uneven branch points in these topologies. Therefore, the mean
of Colless's imbalance measure is higher in the PDA model than in the transient
crBD model.
It
has been found
in many studies
(see [22] and references therein)
that the actual imbalance of real phylogenetic trees lies somewhere between
these
two classes of models.
llt
is for this, and other reasons that we had to define the various mappings, topologies and
probability distributions in Section 3.6.
CHAPTER
3.
MODELS OF MACROEVOLU"IO¡\I
59
The transient crBD model and the PDA model have been the most extensively
studied in the literature. However, the fact that they predict mean imbalances,
based on Colless's measure,
that are too low and too high with respect to real
phylogenetic trees, respectively, [10,
Il,
72, 22, 30] suggests that these models are
not adequate probability measures for macroevolutionary studies.
3.11 Multi-Rate Evolutionary Model
Recall that in 122]
it
was stated
that "most biological taxa
have arisen by a branch-
ing process of descent with modification", suggesting that a multi-type branching
process should be used
to generate phylogenetic trees. Aldous [1] also
proposed
that the multi-type branching process could be used as a model. In fact, since it
is
generally believed that "ercept for mass ert'inct'ions and thezr aftermath, the ouerall
number of speci,es do not tend to'increase or decrease erponenti,allE
fast" [1], ct-
MMTBPs that are close to criticality might make reasonable models. Pinelis
proposed a model called the multi-rate (MR) model and used
[26]
it to show that the
crBD and PDA models, under some fairly stringent conditions, are sub-classes of
the MR model. The MR model is a binary-branch point continuous-time Markovian
multi-type branching process [2, 9] with some slight modifications that we
discuss
below.
More formally, Pinelis [26] considers a phase space S
ç
Z+ where each
is considered to represent a phase or a state that a species can be
¿
eS
in. The phases
in S have the capacity to contain any amount of information, for example, size,
genotype, geographical location, and behavioural patterns of the species. The tree
evolves in the following qualitative manner.
In the interval t to t
I dt arty species
in phase'd €,S may,
1. with probability F1dt transform into another state
2. with probability
o¿¡d,t,
j,
or
remain unchanged and give rise to one new species in
CHAPTER
3.
MODELS OF MACROEVOLUTION
60
phase 7, or finally,
3. with probability
1
- Dr."(F4 -l o¿¡)d,ú it does not undergo any change
The transition rates F4 and o¿¡ for aII
'i,
j
€
and are referred to
.S are non-negative
as the transformation and speciation rates. The rates are assumed
to be constant
in time. Any species in pahse ¿ e S at time ú is evolving independently of all the
existing species and of its history.
The flexibility of the multi-rate evolutionary model stems from the flexibility in
modelling the state space and the transition rate structure. Pinelis [26] proposes one
possible partition of the phase space S into three subsets, D,
D
can be thought of as consisting of those species
U
and Q. Here the set
that are extinct, in other words
С.rjtø -l oor) : 0. In addition the probability
that a speciation event from any species in phase i € E to a species in phase d e D,
o¿¿ is zero. The set Z consists of the species that are classified as unstable, that
for every species with phase d e D,
is they have the capacity to transform and speciate. Thus
for at least one
species
q
if i € U then
j € S. Finally, the set Q is the set of quasi-stable
phases,
)
0
that
is,
ø¿¡
in one of these phases cannot speciate but are also not extinct. Hence, for
e Q, oqi :
0 for all ù €
S, and p,q¿ : 0 for
aII i,
e U UD. As a result,
quasi-
stable phases have the capacity to transform only to other quasi-stable phases. The
quasi-stable phases are interpreted [26] as representing species
that are the most
adaptable. These species "wander" [26] the space of quasi-stable phases, changing
their attributes in order to suite their current situation. For example, they may be
changing their size, feeding patterns, and possibly their genotype to some extent in
order to adapt to a world that is changing around them.
At this point
a note must be made about the position of the parental and daugh-
ter branches at a branch point. Pinelis [26, Appendix A] has chosen the left branch
to be the parental branch and the right branch to be the daughter branch. In contrast, we have chosen throughout this entire thesis to represent the daughter branch
as
the left branch and the parental branch as the right branch. This designation is
CHAPTER
3.
MODELS OF MACROEVOLUTION
61
entirely arbitrary. However, we chose it ahead of the orientation employed by Pinelis
[26] since
it
is a more natural choice when the Markovian binary tree is introduced
and analysed in Chapters 5, 6 and
7.
3.11.1 The PDA Model as an MR Model
It is easy to see that the crBD model is a special case of the MR model with two
states, the extinct state and the unstable state. What is less clear is whether the
PDA model is also a special case of the MR model. Pinelis shows that the PDA
model is indeed a special case of the MR model [26, Appendix A] by makirrg some
fairly stringent assumptions on his model in order to collapse it to a three phase
model. However, the analysis of Pinelis [26, Appendix A] can be performed by
assuming from the beginning that
o
it
is a three phase process, such that:
Phase 0 is the phase of all extinct branches,
o Phase 1 is the phase of all quasi-stable branches, and
o
Phase 2 is the phase of all unstable branches
The phases do not serve as markers for a species as in the approach of Pinelis [26,
Appendix A], but rather they represent the state that a branch may be in at any
particular time.
The analysis below can be found in Pinelis [26, Appendix A]. The transition
rates for the process are:
o transformation from phase 2 to
with rate
phase 0 (represented as
2
---+
0) which occurs
d,
o transformation from phase 2 to phase 7 (2 --+ 1) which occurs with rate
o
a
q,
birth from phase 2 with the daughter branch in phase 1 whilst the parental
branch remains in phase 2 (represented as
b1, and
, --
7,2) which occurs with rate
CHAPTER
3,
MODELS OF MACROEVOLUTION
62
o a birth from phase 2 generating a daughter branch in phase 2 whilst the
parental branch remains in phase 2 (2 -+ 2,2) occ;rtrs with rate ó(1 - 7),
where ?
€ [0,1] and d+b+e:1.
To generate an evolutionary model of the PDA assume that the process is sub-
critical and 1
: ¡.
We consider the mapping, Mø, that maps the quasi-stable
portion of a tree to some topology, Tq in the limit as f
---+
oo. Thus the trees that
we study are those that exist on the space of trees that are finite, almost surely, and
consist of only quasi-stable branches.
The probability that a tree will eventually become extinct is given by,
p(Ø):d+be(Ø)2.
(3.11.1)
Equation (3.11.1) has the obvious interpretation, a tree will become extinct either
directly which occurs with probabllity dl@+b+q)
:
d or via a birth followed by the
independent eventual extinction of the two subtrees and this occurs with probability
bp(Ø)'. Solving for p(Ø), we obtain
P(Ø)
:1 - \42b-m '
'
(3'71'2)
since the other root is clearly greater than one. The equation that characterises the
probability of a random tree obtaining a topology 7q under the MR-PDA model is
p(Tn)
:
up(T¡[,01)n(T¡3,r)
+
2bp(Ø)p(Tq),
(3.11.3)
the daughter and parental subtrees at node [0]. There
are two ways of evolving to a tree with topology Tq. The first term in equation
where
7¡f;,01
and
T¡3,r1 are
(3.11.3) represents the pathway where there is a birth at node [0] and the daughter
subtree evolves to topology Tå,ol and the parental subtree evolves to topologV
T¡3,r,7
independently. The second term in equation (3.11.3) represents the pathway where
a
birth occurs at node [0] followed by the eventual extinction of either the parental
or daughter subtree whilst the other evoives into a tree that has topolo gy Tø.
Pte-
arranging equation (3.11.3) we obtain,
p(Tn):
b
t
-
2be(Ø)
T¡3,0)r6¡3,¡)
(3.11 4)
CHAPTER
3.
MODELS OF MACROEVOLUTTON
It is worth taking
some time
63
to interpret equation (3.11.4) here. Because we have
pruned all dead branches/subtrees, a branch point in which either the daughter
or the parent subtrees becomes extinct is not treated as an actual branch point.
Instead an actual branch point is one in which both the daughter and parental
subtrees consist of at least one quasi-stable branch. The factor l lQ
-
2bp(Ø)) in
equation (3.11.4) reflects this, and is interpreted as giving the expected number of
false branch points before the first actual branch point occurs.
induction that for a tree of topology Tø
wíthlfnl:
It can be shown using
s, equation (3.11.4) becomes,
ps\: (-+@)"-' {r{,))",
where p(1) is the probability
(3
li b)
that a tree consists of a sole quasi-stable branch. Now
p(1) given by,
p(1)
: q+2bp(Ø)p(t),
since a tree that is of size one occurs
a quasi-stable branch, or
(3.11.6)
if the parent branch directly transforms into
if after a birth either the daughter or parental subtree
eventually becomes extinct and the other subtree is itself of size one. Equation
(3.11.6), can be re-arranged to obtain as expected
p(1)
: =--1t - zbe@)
(3.11.7)
The PDA distribution can then be recovered from equation (3.11.5) by conditioning
on the tree size since all topologies of size s are equiprobable. The number of
topologies of a given size, s, can be obtained using equation (3.10.7).
Example 3 Si,ze.four trees
For size four trees there are 5 distinct topologies, four of which belong to topologi-
cally isomorphic class
JF$,n
and one of which belongs to class ß.f,n. For a given size,
CHAPTER
3.
MODELS OF MACROEVOLUTION
64
equation (3.11.5) tells us that all topologies of that size are equally likely. Therefore,
pt(FT,")
Ð"n.
p(Tn)
Drn
¿ol-op(Ts)
-'t-t4p(ro)
5p( Ts)
4
(3.11 s)
5
A similar calculation shows that, p(F'f,nl" : 4) : Il5. More generally then,
the
probability of obtaining a topology from a particular topologically isomorphic class,
IFfl,", conditional on size s trees is,
p"(F'1,")
D"n e- t¡'96,S p(rr)
Ð", ,1"'l:,P(Tn)
2'rn
(3.11.e)
¡/"
In the next section the super-PDA (sPDA) model is discussed. In the sPDA model,
random topologies with higher imbalances occur with higher probability.
3.tL.2 The super-PDA
Model
The purpose of this section is to create a model, similar to the PDA, which gives
higher mean imbalances than the PDA. The model does this by giving higher weighting to unitary branch points, where a unitary branch point is a branch point in which
the daughter subtree is of size 1. In order to define the sPDA model we use the
MR model [26, Appendix A]. The MR formulation of the sPDA model, like the MR
formulation of the PDA model, is a measure on the asymptotic trees that consist of
only quasi-stable branches and the associated mapping
Mq.
The equation for the
probability that a tree eventually has a topology Tq of size s is given by,
p(ro):
b(t - t)
('.O_fu.)'1 I-2b(7-t)p(
where the factor,
factor
=ffir-,
(t * nj-),
i, tf," weighting
)"-' {r{r))",
(3.11.10)
given to unitary branch points, the
is the probability of a branch point, and p(1) is the probability
CHAPTER
3.
MODELS OF MACROEVOLUTION
65
for a size 1 tree. We shall explain these terms in more detail in what follows. The
dependence of the probability on the number of unitary branch points can be thus
clearly seen. Thus when conditioning on the size of tree topologies, topologies with
a higher number of unitary branch points have a higher likelihood of occurrence,
with the completely unbalanced topology having the highest probability. It is for
this reason that the mean of Colless's Index of imbalance is greater in the sPDA
than in the PDA.
We now give the MR formulation of the sPDA model as given
in
[26, Appendix
A]. In the MR formulation of the PDA model an unstable branch is unable to give
birth to a branch that is born quasi-stable; this is reflected by the parameter
j :0.
However, in the MR formulation of the sPDA model the parameter 7 is now nonzero and so daughter branches can now be born quasi-stable. In the MR model for
the sPDA (MR-sPDA), the probability of obtaining an extinct tree is given by
p(Ø):d+b(t-.ùp2(U,
(3.11.1 1)
and has a similar interpretation to equation (3.11.1). The equation for the proba-
bility of obtaining a tree of topology Tq is given by
p(Tn):
b(1 -r¡r(T¡3,0)rØ¡3,r1)+2bp(Tq)p(Ø)+heØó,,1)r{17¡3,orl:
t},
(3.11.12)
where once again, Tt\,ol and T¡\,r¡are the parent and daughter subtrees at node
and
1{lff\,oll:1}
0
is the indicator function of the event that the daughter topology
consists of a single quasi-stable branch.
In the MR-sPDA model, a random tree
of topology Tø has three possible routes from which
it
can evolve. The first is
via a non-unitary branch point at [0] such that the daughter and parental subtrees
subsequently evolve into topologies 7¡fi,01 und T¡S,tl independently; the first term of
equation (3.11.12) reflects this route. The second is via a non-unitary branch point
at [0] such that either the daughter or the parental subtree
eventually becomes
extinct, whilst the other evolves into a tree of topology Ts, tlnis is reflected by the
second term of equation (3.11.12). The
third and flnal route is via a unitary branch
CHAPTER
3.
MODELS OF MACROEVOLUTION
66
point at [0] such that the parental subtree then evolves to a tree with topology
7,fr,r,, this is reflected in term three of equation (3.11.12). However, the third term
is non-zero only
if the daughter subtree at node
[0] of the random tree consists of
only one quasi-stable branch. Equation (3.11.12) can be re-arranged to obtain,
b(!-t), ,,,*
n(Tq\:
r,\) - (\r_zair_ùp(Ø),
r{lrfi,ql :
1+
tl
)
.Y
(r-z)p(lrr$,ql:t)
r{lrfi,ql : t}
n6¡3,0)nV¡3,a)
PeQ¡[,q)n(rr\,,),
(3.11.13)
where {l
: (A¡ -
7))
2b(I
l0 -
-
ùp(Ø)). Equation (3.11.13) can also be written
in another way, namely,
p(To) : ( r+
\
where p(1)
:
p(lTó,ql
ltt-?i8'o
,--1
(1 - t)p1)
^ )¡
: t)
t--"
pr(r3,o)n(T¡3,r7),
(3.11.14)
is the probability that a subtree consists of a single
quasi-stable branch. The probability of a one branch subtree is
p(1)
: q+2b(t-t)p!)p(Ø)+he@).
(3.11.1b)
This equation can be simply understood by noticing that a branch may become
quasi-stable either directly, with probability q, or a birth may occur and then either
the daughter or parental branch evolves into an extinct subtree whereas the other
eventually becomes quasi-stable, and this happens with probability Zb(1-1)e(I)p(Ø),
since either the daughter or parent may become extinct, and finally, a birth may
occur such that the daughter is born quasi-stable and the parent subsequently becomes extinct, which occurs
with probabtlity, fup(Ø). W" can re-arrange equation
(3.11.15) to obtain
q + hp9)
p(r): 1-2b(I-t)p(
(3.11.16)
)
This equation can be interpreted in a similar way to equation (3.11.4). Once again
the effect of pruning all the extinct branches and hence subtrees at a branch point
CHAPTER 3.. MODELS OF MACROEVOLUTION
means
case,
0/
that we do not treat such branch points as actual branch points. In this
what is occurring is that at each false branch point either the parental branch
or the daughter branch is pruned, Ieaving us with one unstable branch. Since the
process is sub-critical the unstable branch must eventually undergo either a direct
transition to become quasi-stable or alternatively undergo a false branch point such
that the parental branch becomes extinct and the daughter is born quasi-stable. The
factor 1l(1 -2b(7 - ùp(Ø)) reflects the fact that this occuïs and is interpreted as
being the expected number of false branch points before a transition that generates
a quasi-stable branch occurs.
As with the PDA model one can use induction to show that equation (3.11.14)
becomes
p(rn):
(t. FfuO)" o"-'(r(1))",
(8.11.12)
if the random tree is of size s, and where u denotes the number of unitary splits
in the topology Tq. Equation (3.11.17) demonstrates the sPDA property, that is,
that topologies with a larger number of unitary splits have a higher probability of
occurrence.
The purpose of this chapter has been to describe an alternative state space for
branching processes, a state space that lends itself to macroevolutionary modelling.
We have also described a number of measures from the space of trees to the space
of tree topologies and their associated mappings. \Me then followed this by detailing
one of the most used measures of imbalance
in phylogenetics
research, Colless's
Index. We then began to analyze a number of well known models, the crBD and
the PDA model in particular. This chapter was concluded by an analysis of the
most sophisticated model to date, the MR model of Pinelis, [26]. This provides the
background for beginning our introduction and analysis of the Markovian binary
tree, the model that we propose which is equally sophisticated and yet more versatile.
To begin to understand the language with which we define the MBT, we need to
first discuss some further background materiai. This is done in Chapter 4 where
some important concepts
in the area of Matrix-Analytic methods are introduced.
CHAPTER
3.
MODELS OF MACROEVOLUTION
68
This then allows us in Chapter 5 to define the Markovian binary tree (MBT). In
that chapter we consider the modelling flexibility of the MBT. In particular,
we
show that the PDA, sPDA and the general MR model can be written in terms of
the MBT model.
Chapter
4
Matrix Analytic Methods: an
Introduction
4.L Introduction
The theory of matrix analytic methods forms the foundation of much of the work
that is to follow in this thesis. We represent the binary-branch point continuous-time
Markovian multi-type branching process (ctMMTBP) as a level-dependent quasibirth-and-death process (QBD) and call
it the Markovian binary tree (MBT).
This
representation, in contrast to the classical branching process representatiorr, opens
the door to the possibility of performing efficient numerical analysis and obtaining some useful measures with which to model phenomena such as macroevolution.
In order to motivate the QBD we begin by discussing one of the simplest of all
stochastic processes, the Poisson process. The Poisson process, with exponential
inter-event times, has enjoyed enormous popularity
in applied probability due to
its simplicity and wide applicability. However, there are times where such a process is just not flexible enough to provide a useful model. Therefore, the need to
develop more compiex stochastic models that still retain a certain degree of the
mathematical elegance of the exponential distribution, led to the development of
69
CHAPTER
4. MATRIX ANALYTIC
METHODS: AN
INTRODUCTION
70
the phase-type distribution (PH) The PH-distribution was utilised to generate the
phase-type renewal process from which the more general Markovian arrival process
(MAP) was spaluned. To obtain the MBT representation of the binary-branch point
cIMMTBP, we embed the MAP into the branching process in order to generate
a
phase process on each living branch of the MBT. This allows us to re-interpret the
transition structure of the binary-branch point cIMMTBP, a structure that is devoid
of correlations between particle lifetime and the types of offspring that are spawned,
to a transition structure that has the flexibility of allowing such correlations. This
re-interpretation is based on distinguishing between transitions that are observed
and transitions that are hidden.
The Poisson process has played a central role in modelling real world phenomena
for many years. For example,
it can be used to model radioactive
arrival of customers to a queue. Let
X(t)
decay, or the
denote the random variable that counts
the number of events that occur in a time interval (0,t], where the arrivals follow a
Poisson process. The state space for
popularity to the fact that
it
X(.) is {0} ¿Z+. The Poisson process owes its
possesses a number of useful properties:
1. For all ús : 0 ( úr 1 t2 {
..
.1
tn<
.. ., the random variables X(t"+t)-X(t")
are independent,
2. the time interval between two
successive events is exponentially distributed,
and
3. the random variables
X(t+ s)- X(s)
only on the interval length,
have a Poisson distribution and depend
ú.
The probability that an event occurs at a time s
P[" <
(
ú
is given by,
t]:t -exp(-lú),
(4 1.1)
where 1/À is the mean of the process. The exponential distribution has probability
density function given by
p(t): Àexp(-lú).
(4.1.2)
CHAPTER
4.
MATRIX ANALYTIC METHODS: Alú
Furthermore,
that
it is easy to show that
INTROD'UCTION
TI
the exponential distribution is memoryless,
is,
P[t < úf
ú6ls
>
¿o]
:
P[s <
f].
(4.1.3)
The memoryless property lies at the heart of Markov processes and explains their
SUCCESS
Section 4.2 studies the phase-type renewal process.
In Section 4.3 the MAP
is analysed and the correlations that may develop within the process are emphasised. Section 4.4 discusses the level-independent QBD and Section 4.5 discusses
some algorithms
that are of importance: the algorithm of Neuts, algorithm U and
the level-independent logarithmic reduction algorithm. The first two of these algorithms will be adapted to the MBT
in Chapter 7.
Section 4.6 introduces the
level-dependent QBD. Finally, Section 4.7 discusses the level-dependent iogarithmic
reduction algorithm.
4.2
Phase-Type Renewal Processes
The ease with which the exponential distribution can be applied prompted a search
for a generalisation that still retained many of its useful properties. The phase-type
(PH) distribution provides the matrix generalisation of the exponential distribution.
A phase type random variable is the time to absorption in state 0 of a Markov process
on the phases {0, 1,.
..,n}
and a phase type distribution is the distribution of such
a random variable. An excellent introduction is given
in
[1S] and [23].
The initial probability vector of the process is given by (ro,z) with
ro+Te:
Lj
whereeisavectorofonesoftheappropriatedimensionand¡isalxnnon-negative
vector. The infinitesimal generator is given by
Q:
where d is an n x 1 vector and D6 is an
00
dDo
(4.2.1)
nxn matrix. Because Q is the infinitesimal
generator matrix, we require that the diagonal elements of Ds, (Do)oo, be strictly
CHAPTER
4. MATRIX ANALYTIC
negative for all e e {1,
...,n}
METHODS:
Al{ INTRODUCTIO¡\I
}
and that d¿ > 0 and (D6)¿¡
0 for 7
72
<iI j <
n
Finally, we require that the process is conservative, that is,
Dse
I d,:
(4.2.2)
O
As can be seen from the form of Q, once the Markov process enters phase 0 it can
never exit phase 0.
Definition L
The d'istributi,on of ti,me X (t) unti,l the process ts absorbed i,n the state
0 i,s a phase-type di,stri,but'ion and i,s denoted by P H (r , Ds) .
P
(r,D6) has distribution function,
H
F(r):7-rexp(Dsn)e.
It
is clear then, that equation (4.2.3) provides the generalisation of the exponential
distribution as given in equation (a.1.1).
phase 0 occurs from any phase i,
e {7,.
..
It is shown in [18] that absorption into
,n} with probability 1 if and only if the
matrix Ds is non-singular, that is, if Do is invertible. Then
total time spent in phase
of
(4.2.3)
j
(-
Do)øt is the expected
during the time until absorption, given an initial phase
i,.
Consider the Markov process defined above that commences at time úo
a phase determined by the distribution
¡.
:
0 in
Instantaneously upon absorption at t1,
restart the Markov process by choosing a new phase from that same distribution,
T.
The process then proceeds until absorption, which now occurs at time t2 and
from here restart
it
ts <, t7 1 t2 1
..
again according to that same distribution
.Ì
z.
The set {0
:
of reinitialised time points forms a renewal process. The
inter-renewal distribution is given by PH(r,D6). We call this process a phase-type
renewal process.
Let the above Markov process be denoted fV {óQ)l¿ > O}, where ó(t) e {7,2,.
..
,n}
We call this Markov process the phase process. The infinitesimal generator of the
phase process is given by,
D
:
Do
I d,.r.
(4.2.4)
CHAPTER
4. MATRIX ANALYTIC
METHODS: A,Iü
INTRODUCTTON
73
The form of D indicates that there are two ways in which the process can move from
phase z to phase
1. directly,
j,
i,- j
and they are, either
which occurs with probability (D6)¿, f (-Ds)¿¿, or
2. indirectly, via absorption in 0 from
z and then being immediately restarted
phase 7, which occurs with probability
in
d¡r¡leDs)¿i.
Let ,n/(ú) be the random variable that denotes the number of renewals that have
occurred up
to and including time t. The two-dimensional representation of
phase-type renewal process is given by {(¡r/(¿) ,ó(t))l¿
state space, {0} U Z+
the
> O} which is defined on the
x {7,2,. . . ,n.}. The Q-matrix for the entire process is
Do d.r
0
0 Do d.r
00Ds
Q:
(4.2.5)
The process can be partitioned into levels: if I renewals have occurred then the
process is said to be in level
l, L(l), and the set of states at each level
L(l):
for all
l>
{(¿, 1),
are
(1,2),...,(l,r)},,
1.
The process is homogeneous with respect to ievel, so that the future evolution
of the process does not depend on its current level.
The probability generating function of the number of events up to time ú in
phase-type renewal process is [18],
P(z,t)
-
exp ((¿o +
zd.t)t),
and the corresponding probability generating function for the Poisson process is
P(z,t)
-
exp ((z
-
1)^r)
a
CHAPTER
4. MATRIX ANALYTIC
METHODS: Alü
INTRODUCTION
74
As a result the phase type renewal-process can be thought of as providing the matrix
generalisation of the Poisson process. The exponential inter-renewal distribution of
the Poisson process has been replaced by the phase-type distribution of the phasetype renewal process.
In Section 4.3 we generalise the phase-type
renewal process
to the Markovian
arrival process (MAP). In particular, we shall study the type of MAP called the
transient MAP [19]. Transient MAPs, unlike ordinary MAPs, terminate after
a
finite number of arrivals, almost surely. For the remainder of this chapter and thesis
we shall be dealing
with transient MAPs and referring to them genericallv
as MAPs.
4.3 Markovian Arrival Processes
The core process that governs the growth of a Markovian binary tree (MBT)
is
the Markovian arrival process (MAP) 120, 251. The MAP is a continuous time
Markov process with two dimensional representation {(,V(t),ó(t)): ú € IR+}, on
the state space Z+ x {0,
1, . . .
,n},
where
r¿
is a finite integer. There are two types
of transitions: hidden transitions and observable transitions. The random variable
,n/(ú) counts the number of observable transitions
ú. The random variable
@(ú),
that have occurred up to time
which denotes the phase of the process, evolves as
a
continuous-time Markov chain.
MAPs that have almost-surely infinitely-many points have been studied for
a
long
time and were first introduced by Neuts [25]. A later paper [20] introduced a more
economical notation and studied a number of other properties. In [19], Latouche,
Remiche and Taylor extended the concept so that a MAP can have almost-surely
finitely-many points, known as the transient MAP. These transient MAPs cease to
evolve at some catastrophe time
use
to generate MBTs.
T. It
is the class of transient MAPs that we shall
CHAPTER
4. MATRIX ANALYTIC
METHODS; Alü
INTRODUCTION
T5
The transition rate matrix for the MAP is
Dö Di
ODð
00
Di0
00
00
Dö Di
Q:
where,
ODð
oll,andDi:ll-o oll,
Dä: [o
I
ld no)
and the matrices Do,
(Do)o¡
>
0
if i,+
j,
Dt
and
d
lo o')
have the properties
that, for aII i,,j
: 7,...,fr,
< 0, (Dr)oj > 0 and d,¿ > 0 with at least one k for which
that the matrix Dol Dt is irreducible. If this is not the case
(Dr)no
dn> 0. It is assumed
then there are redundant phases.
Using e to denote a vector of ones of the appropriate dimension, the matrices
Dfi and Df are such that,
D[e +
Dle:
O,
which is equivalent to
D¡e
I
Dte
I d,:
O.
The z-th entry of the vector d contains the rate at which the process ceases to evolve
when the phase process is in phase i. In the traditional formulation of the MAP,
due to Neuts 125],
d:
O and phase 0 is removed.
Remiche and Taylor [19] have
d
)
The transient MAPs of Latouche,
0 componentwise with at least one k for which
d*>0.
Let (d¡)¿ : -(Do)¿¡ for all i, e {I,2,. ..,,n}. Suppose that the current state of
a MAP is (rn, i,), that is, there have been rn observable transitions and the phase
process is
in phase
i.
The process will remain in this state for an exponentially
distributed period of time with mean 1l@ù0. At the next transition there are three
possibilities:
CHAPTER
4. MATRIX ANALYTIC
o for j I i,, wilh probability
METHODS; AIü INTRODUCTION
(Do)o¡l@o)¿ there
phase 7 and so the new state is (m,
76
will be a hidden transition to
j).
o With probability (Dt)o¡ l@o)¿ there will be an observable transition to
7, so that the new state is (m+7, j).
phase
o Finally, with probability dol@o)o there will be a transition into phase 0. The
new state of the process is then
(^,0).
We call such an event a catastrophe.
We model the occurrence of a catastrophe by saying that the process enters phase
0. Latouche, Remiche and Taylor [19] envisaged that this transition could be either
observable or hidden. However it makes sense in our context for the catastrophic
transition to always be hidden.
MAPs have the interesting property that the phase process can generate correlations between the time between two observable events and the phase of the process
immediately after the second of those observable events. These correlations arise
because the transition rates depend on the underlying phase. What interests us
here, is the distribution of the phase immediately after an observable event. As a
first step, we determine this distribution and then illustrate the correlations through
a concrete example.
The probability (Pn(t))o¡ that, at time ú, the phase of the process is j and the
intervening phase transitions are all hidden given that the process began in phase
z
is given by
P¡(t):
exP(D6Ú)'
The density (R.(t))¿¡ that the first observable event occurs at time ú and changes
the phase into phase
j,
given that the process began in phase d, is given in matrix
form by
R"(t): exp(D6f)D1.
(4.3.1)
The probability, (P"a"(t))¿¡, that the phase of the process is 7 immediately after the
first observable event, given that the phase process began in phase i, and the first
CHAPTER
4. MATRIX ANALYTIC
ME,THODS: AIú
INTRODUCTION
77
observable event occurs at time ú, is
(R"(t))u
),t:ffi
(P",,(t)' -
(4.3.2)
'
Equation (4.3.2) illustrates the fact that the distribution of phase immediately after
the first observable event is dependent on the time
¿
of that first observable event.
Example 4 Four phase MAP
Consider the MAP on phases
{1,2,3,4} defined by
-20.0000
0.0000
0.0000
0.1000
0.1000
-20.0000
19.4000
0.1000
0.1000
19.0000
-20.0000
0.1000
0.1000
0.1000
0.1000
-20.0000
Do:
and,
Dt:
0.0000
0.1000
0.0100
19.3000
0.0000
0.0100
0.1000
0.0000
0.1000
0.2000
0.1000
0.2000
15.0000
0.1000
0.1000
0.1000
An easy calculation shows us that
P,¿"(0.0t)
:
P,o"(O.t;
:
0.0008
0.0052
0.0045
0.7249
0.1849
0.4542 0.2260
0.L766
0.3070
0.1806
0.3357
0.9791
0.0065
0.0065
0.0078
0.9936
0.0077
0.0052
0.0006
0.2295
0.2063
0.1987 0.3655
0.2290
0.2099
0.1946
0.3665
0.9667
0.0069
0.0068
0.0195
and
0.9866
The most striking indication of the dependence of the phase distribution immedi-
ately after the observable event on the time of the event comes when one looks
4. MATRIX ANALYTIC
CHAPTER
at
(P,6"(0.01))r,
(P,a"(0.1))z¿
:
:
METHODS: Al{
0.4542, (P,å"(0.01))rn
0.3655. Thus
if the process
:
INTRODUCTION
0.2260 and (P,6"(0.1))æ
:
78
0.1987,
begins in phase 2, Lhe earlier the first
observable event the more likely that the process
will enter phase 3. In contrast,
the longer the inter-observable event lifetime the more likely that the process will
enter phase 4 at the first observable event. Consider the observable transition from
phase
1to phase L att:0.01
P,a"(O.l)rr
that
:
space of
we have P,a"(O.0t)tr
:0.0008 and at ú:0.1 we have
0.0077, the probability of such an event increases almost 1O-fold in
time. The reason for this is that from phase
1 there cannot be a
direct
observable transition back into phase 1, so as time increases the probability that a
hidden transition say to phase 4 can occur followed by an observable transition from
phase 4 to phase 1 increases in likelihood.
Example 4 illustrates the important property that the lifetime distribution and
the distribution of phase after an observable event are correlated. The level of
correlation depends on the nature of the matrices Do and
Dr
As will be seen in
Chapter 5, by constructing the MBT using a MAP process on each branch, 'we can
model correlations between the initial phase, branch lifetime and the initial phases
of the offspring.
Let P¡¡(t) be the probability that there will be an observable transition to phase
7
within the interval
[0, ¿] given
that the process began in phase
i, and
P(t)
:
lPo¡(t)1.
Then,
P(t)
exp(Dsr)Dñr
(-ro)-'
(1
-
exp(D ot)) Dt
(4.3.3)
Figure 4.3.1 graphs the time-dependent changes to the probability for an observable
transition into the four phases given that the process began in phase 2 for the
MAP from Example 4. This figure demonstrates quite clearly how the probability
of an observable event occurring depends on time. For example, an observable
transition from phase 2 to phase 3 is the most probable up until approximately time
ú:
0.06 at which point phase 4 becomes the most probable. Furthermore, up until
4. MATRIX ANALYTIC
CHAPTER
approximately
t:0.11
METHODS: Alü
INTRODUCTION
T9
an observable transition into phase 1is the least probable
transition, because the transition2
of hidden transitions, say a 2
---+
---+
1
cannot occur directly but requires a number
4 transition followed by a 4 --+ 1 transition. The
probability of these transitions increases over time thus increasing the probability of
a2
---+
1 observable transition; the observable 2
---+
7
transition becomes the second
most likely observable transition as time increases. These observed behaviours are
due to the complex internal hidden transitions that are generated by the D6 matrix.
004
Probability Distribution for an Observable Event against Time from phase 2
-+-
*
€
0.035
Phase 1
Phase 2
Phase 3
Phase 4
p'
0.03
,D
ø
ø
Ø
ø
ø
0.025
ø
,l
.*
¿l
t
d o02
¡o
(I
0.015
ø
P
*
+ -{
+' +
ø
,l
0.01
* *
Ø
0.00s
*ø
0
01
0.05
0.15
o2
o.25
Tme
Figure 4.3.7: Probability distribution for an observable event against time given
that the process began in phase
It
2
is clear from equation (4.3.3) that the time until the next observable transition
is not exponentially distributed. Let
7
be the mean time until an observable event
CHAPTER
4. MATRIX ANALYTIC METH)DS:
Alü
INTR)DUCTI)N
80
occurs given that the process began in phase z, then
T-
I,*
ú
exp(Dsú)
Dpdt
(4.3.4)
After integration by parts this can be shown to be
- -
(-Do)-"
Dr".
(4.3.b)
to Example 4, the mean time between observable transitions
conditional on the initial phase is,
Once again referring
0.0492
T-
0.8451
(436)
0.8367
0.0475
This is what we would have expected since phases 1 and 4 have a high probability of
triggering observable transitions at earlier times, whereas phases 2 and 3 are more
likely to undergo more hidden transitions before an observable one.
Having introduced the MAP we now are ready to begin a brief exposition on
quasi-birth-and-death processes (QBDs). The importance of the QBD in this work
is that its provides an alternative framework to that of the branching process frame-
work within which to define the Markovian binary tree. In particular, it will be seen
that despite the fact that the QBD and the traditional branching process representations of the MBT refer to the same underlying processT the QBD interpretation
confers to a process otherwise devoid of interesting lifetime-offspring correlations a
much richer structure.
4.4
Level rndependent Quasi-Birth-and-Death proCESSES
The level-independent QBD process is a Markov process,
the two dimensional state space
{x(t)lt e 101 UIR+} on
{(^,i)lnz e {o} uzt,r < i, < n}. If the process
CHAPTER
4. MATRIX ANALYTIC
METHODS: AIú INTRODUCTION
81
(^,i) at some time ú, we say that the process is currently in level L(^)
and in phase i. The state space is therefore partitioned into levels, L(*), where,
L(^): {(m,7),(^,2),...,(^,n)} for all m} 0. Consequently, for ail rn there are
is in state
exactly n states in L(m). The infinitesimal generator of the process is given by
8:
where
00
BAo
Az Ar
042
00
B, Ao, A1 and A2 are aII n x
r¿
are non-negative except for (A1)¿¡ and
Aoo
Ar
(4.4.t)
Ao
A2 AI
matrices. All the elements of the matrices
B¿¿
which are strictly negative for all ri e
{7,2,. . .,n}. In addition, the matrices Ao, A, and A2 obey (A2 t At 1- A')e :
0 where e is a vector of ones of the appropriate dimension. It is assumed that
Az
I
At
f
.40 is irreducible.
Suppose that the process is currently in state
(^,i).At
the next transition the
process may either
o move down to L(m - 1) bV entering state (rn
o remain in the same level by entering state
-I, j)
with rate (A2)¿¡,or
(^,j), j + i., with rate (A1)¿¡, or
finally,
o move up to L(m + 1) bV entering state (rn + I, j) with rate
(Ao)o¡
In keeping with the traditional literature we call transitions that move down a level,
left transitions and transitions that move up a level, right transitions.
It
is sometimes important to determine the probability of eventually moving
from L(m) down fo L(m - 7). This probability is given by the matrix G. Let
.y(t) : inf{ú > 0lX(ú) e L(()} be the first passage time inro L(m). The probability
that the process eventually reaches L(*
-
1) for the first time and does so in phase
CHAPTER
4.
k, given that
G¿n:
it
MATRIX ANALYTIC METHODS: Alü
s2
commenced in state (m,i,) is
Ph(*-1)
<æ
kX(1(^-1)):(m-r,k)lx(o) :(*,i)1.
Notice that since the process is independent of level,
a similar vein
INTRODUCTION
to
G
G¿¿ does
(4.4.2)
not depend on m. In
[18] we shall adopt a slight abuse of notation and write,
:
Plt(m
- 1) < oo & x(7(rn - r))lx(o) e L(m)1,
(4.4.2)
for equation @.a.2) in order to "avoid storms of subscripts" [18], thereby making the
equations easier to read. Since this process is homogeneous with respect to level,
G is independent of the IeveI, L(m), from which the process commences, provided
that m )
It
0.
can be shown that G is the minimal non-negative solution to the matrix
quadratic equation [24]
F
Since
: (-At)-tAr+ (-Át)-t AoF'.
(4.4.4)
G is the minimal non-negative solution to equation (a.a.Q we often write
(4.4.4) as
G
: (-At)-t
A, + (-At)-t AoG'
.
(4.4.5)
Equation (4.4.5) has a very neat physical'interpretation. If the process begins in
L(*)
- L). The first way is a
direct transition to L(m - 1) which occurs with probability (-Ar)-tA". In the
then there are two ways of moving down to L(m
second way, the process undergoes a right transition and moves up to
probability
from L(m
L(m+
1) with
(-Ar)-tAs. From here the process eventually moves down to L(m_ r)
+ 1) by first movingto L(m)
and then moving to L(m
-
1); each of
these transitions has probability, G. Due to the independence of each of these three
events, the probability of this second way is (-A)-1AoG2.
By repeatedly substituting the left hand side into the right hand side, equation
(4.4.5) can also be expressed as
c:
U, ?
Ë
t:o
Ar)-, A2
:
(r
- u)-, (- Ar)-, Ar,
(
4.4.6)
CHAPTER
where U
4. MATRIX ANALYTIC
: (-At)-'AoG.Equation
METHODS: AIú INTRODUCTION
83
(4.4.6) also has a neat probabilistic interpreta-
tion. The matrix [/ is the probability of first return to L(m) under the taboo that
L(*) given that it began in L(m). Mathematically,
the process does not go below
u:
Pl1(m)
<t(^ -
1) &
[/
Similarly to G, the matrix
Each individual term,
t(m) <
oo &
x(7(m))lx(O) e L(m)1.
does not depend on the
(4.4.7)
initial level.
U'(-At)-rAz, of equation (a.a.Q
gives the probability
that the process will return to L(m) I times, before it eventually undergoes a left
transition and enters
L(^-
of eventually entering L(m
1), given that
-
it
began in
L(m). The total probability
1) is given by the sum of these terms for all l.
In addition to G being the minimal non-negative solution to equation (4.4.4) it
can also be shown [18] that G is the minimal non-negative solution to
oo
r : t (e,qù-',40r)kÇ,+r)-tA,
k:0
A number of important numerical
(see [7,
(4.4.8)
schemes have been developed
to solve for G,
77,16,78,24,23,271). The two of most interest in this work are the Neuts
algorithm which is also called the method of modified substitutions, 124,23] and
algorithm t/ [18]. The Neuts algorithm, which is discussed in Section 4.5.1, is an
iterative scheme based on equation @.a.\ and aigorilhm U, which is discussed in
Section 4.5.2, is an iterative scheme based on equation
will form the basis of algorithms
we develop for the
( .a.S).
These algorithms
MBT and then the more general
Markovian tree in Chapters 7 and 8, respectively.
4.5
Level Independent Algorithms
4.5.L The Algorithm of Neuts
The first algorithm that we discuss here is called the algorithm of Neuts, or the
method of modified substitutions [24,23]. The algorithm is developed by considering
CHAPTER
4. MATRIX ANALYTIC
METHODS: Alü
INTRODUCTION
84
equation (4.4.4),
F: (-At)-'At+ (-Ar)-t AoF'.
(4.5.1)
Neuts 124, 23] showed that the sequence of matrices defined by,
c(o) : (-Ar)-'Az
G(t)
for I
)
(4.5.2)
: (- Ar)-t Az + (- Ar)-r AoG2 çt - t¡,
(4.5.3)
1, are non-decreasing and converge to the minimal non-negative solution, G,
of equation (4.5.1).
Consider the set S¿, which contains all sample paths of the process that begin in
L(*)
and which eventually visit, L(m
is L(m +
-
1), such that the maximum level reached
l) with the added restriction that
each of the sample paths has at most
2¿
left transitions.
Level
3
Level2
Level
I
1
Level0
Path
1
Path 2
Figure 4.5.7: Two sample paths in
52
The Neuts algorithm at iteration I only considers sample paths from the set S¿.
However, the space of sample paths included at the l-th iteration is a strict subset of
this set. To illustrate this, Figure 4.5.1 depicts two sample paths from 52. Sample
path
1 is
included at the step I
:
2 of the Neuts algorithm but sample path 2 is not,
sample path 2 is considered at the next step of the algorithm. The space of sample
paths included at each step of the algorithm is not easily described. We will show
4. MATRIX ANALYTIC
CHAPTER
METHODS: Alú
INTRODUCTION
85
in Chapter 7 a description is possible by a suitable transformation to the space of
binary trees.
4.6.2 Algorithrn
U
Recall that U gives the probability that the process will eventually return to L(m),
given that
it began in L(m), under the taboo that it
does not move down to
L(m-L),
in equation (4.4.7), and G is the probability that the process will
enter L(m - 1) given that it began in L(rn). Algorithm U is defined as,
as expressed
eventually
: (-A,)-'A,
M(t) : (-Ar)-'AoF(¿ - 1)
F(t) : (r - MØ)-'(-A,)-'A,,
F(0)
(4.5 4)
(4.5.5)
1
(4.5.6)
forl)1.
Now consider the sequences {y(¿)} and
u(t): Ph(^)
<.y(rn
{G(l)} for I }
1, defined by
- 1) & t(m) <.y(t+m+7) k x(1(m))lx(O) e L(m)1,
and
G(t):Ph(^-1)
The matrix
taboo that
U
it
<
t(m+l+t) kx(1(m-1))lx(0) eL(m)|
(l) is the probability that the
doesn't visit L(m
-
1) and
it
process
(4.5.7)
will return to L(m) under the
can reach at most
L(m+ l)
given that
it began in L(m), and G(l) is the probability that the process eventually
L(^- 1) and it can reach at most L(m+ l) given that it began in L(m).
enters
Latouche and Ramaswami [18] give a simple, elegant physically-motivated proof
that shows that the
verge
sequences
U(l) and G(l) are monotonically increasing and
con-
to the matrices U and G respectively and that G(l) and U(l) are identical
to ,F(l) and M(l) for all ¿ > 0 respectively. They also show that G is the minimal
non-negative solution to equation (a.a.8).
CHAPTER
4. MATRIX ANALYTIC
METHODS: Alú
INTRODUCTION
Algorithm U converges linearly with respect to level.
It
converges
86
at a faster
rate than the Neuts algorithm, because at the l-th iteration of algorithm
[/
all the
sample paths from the l-th iteration of the Neuts algorithm are included, in addition
to many more, because algorithm U places no restriction on the number or pattern
of left transitions.
4.5.3 The Level-Independent Logarithmic Reduction Algorithm
The space of sample paths that are measured at each step of algorithm
[/
increase
linearly with respect to the taboo level. Thus at the l-th step the sample paths
consist of those paths that commence at
L(^)
and end in
L(m-
1) under the taboo
that the maximum level reachedis L(m-ll) without placing any other restrictions on
the number or positions of the left or right transitions. An algorithm that converges
quadratically with respect to level was developed by Latouche and Ramaswami
[17]. This algorithm is called the level-independent logarithmic reduction algorithm
(LILRA).
It converges quadratically because
the maximum level that the sample
paths can reach does not increase linearly per step but geometrically.
Consider the matrix, HUl, whose definition is,
¡7u): ph@+2,) <.y7n-2,) k x(t@¡2,))lx(o) e L(m)1,
for m
>
2¿.
The matrix.Fl[¿] has a simple physical interpretation: it is the probability
that the process will enter level L(m+
in L(m). The matrix, LU), is defined
2¿) befor e
L(m-
2¿) given
The physical interpretation o1 ¡ltl is very similar ¡o ¡[lt):
the process reaches
L(* -
L(m). The
2¿)
that it commenced
as follows,
¡ut: pd@-2,) <t(m+2,) t x(t@-2,))lx(o)
began in
(4.b.s)
L(m)1.
(4.b.e)
is the probability that
L(m* 2¿) given that the process
so it does not matter from which
before ever entering
process is level-independent
it
e
CHAPTER
4. MATRIX ANALYTIC
METHODS:
Ievel, L(m) we commence provided that m
(4.5.s) and (4.5.9)
¡yul
)
Al{ INTRODUCTION
21. We can therefore
BT
write equations
as,
: phe,*') < ?(0) tz x(tel+'))lx(o)
e Le\1,
(4.5.10)
and,
¡ut
:
plry(o) <.y(2,*,) &
Define the matrix ¿¡[t)
process returns
6
x(7(0))lx(o) e L(2t)].
(4.b.11)
be the probability that commencing from L(2t+1) the
to that level after visiting L(Zt+t
i
2t) or L(2t) but before visiting
L(Zt+27 or ^C(0) and is given by,
¡¡U)
for I
)
:
gltJ
¡tt) ¡ tr[t) ¡¡It) ,
(4.5.r2)
0. Let us commence from 4(1) so G is,
G:
Pfi(0) <
oo &
x(7(o))lx(o) € ¿(1)1,
and can be determined using the level-independent logarithmic reduction algorithm
[17] that is stated below.
Theorem
8
The matrir G ,is gi,uen by,
G
t
glt)
¿>0
(4.5.13)
1
where,
¡7tol
trlo)
glt+t7
trlt+t)
forl>0
: (-Ar)-t Ao,
: (-Ar)-'Ar,
:
:
(r
-
(4.5.14)
(4.5.15)
uut¡-, (nut¡'
(I - Ultl¡-'lfrrr'
,
,
(4.5.16)
(4.5.t7)
CHAPTER
4. MATRIX ANALYTIC
METHODS; AIú
INTRODUCTION
88
The proof can be found in [17] and [1S]. The l-th term in (a.5.13) contains all those
sample paths that commence in 4(1) under the taboo that they never visit L(21+1)
before entering
4(0). This is a powerful
approach when one considers that at the
l-th step of aigorithm U only those sample paths that reach at most L(l + 1) are
included.
The remainder of this chapter is dedicated to the level-dependent QBD (LDQBD)
4.6
Level-Dependent Quasi-Birth-and-Death ProCCSSCS
A level-dependent QBD (LDQBD) is defined in the following manner [3]: let X(ú)
be a two-dimensional Markov process on the state
o<
M^
Q e {1,
)
(*,a)
{
:
If X(t) (m,tÞ) we say that the QBD is in L(m)
...,M^}.
M^ distinct
rn
0
and
1
in phase
The number of states of the process in each level is given by the
number of phases in each level, so if there are
are
space
M-
phases at level
L(^)
then there
states.
The infinitesimal generator of the LDQBD is,
0Í')
a[')
00
atÐ e\') 8L')
8:
o
00
where the matrix QY)
M^
X
Ql:)
a?)
o
(4.6.1)
atr)
af) al')
i" of dimensi on M^ x Mm-t, Q?) ¡" M^ x M^
Mm+r. The entries in each matrix for m
)
and
S[-) it
0 are strictly non-negative, except
) 0. In
we assume that the process is irreducible and fhat Qf) e¡qf) e+QP) :
the diagonal elements of Ql-) which are strictly less than zero for all m
addition,
O where each
"
e is of the appropriate dimension.
CHAPTER
4.
MATRIX ANALYTIC METH)DS: Alú
INTR?DUCTI2N
89
The equation for the family of matrices, {G^,rn > r} is slightly more complicated in the level dependent domain. The probability of eventually entering state
(^-7,7)
(G*)o¡
starting from (m,z) is (G-)¿¡, where
: plt(*-
1) < æ
The family of matrices
k x (t(-
- r)) :
(m
- r, j) lx(o) :
{G^,m > l} is the minimal
(m,i.)1.
(4.6.2)
non-negative solution
to the
family of equations,
G^: eeY))-,q?) + eAY))-reP)G*+rG*,
) 7. Physically
ror m
then, the process may move down to L(m
directly with probability
ability
eqy))-'Q[-)
eQy))-tqf)
ot by first moving to
each of these events is independent we have
4.7
L(m*
1) from
L(-)
1) with prob-
ana then eventually moving down to L(m) with probability
Gm+r, and then eventually moving down to L(m
approach
-
(4.6.3)
-
1) with probability
G-.
Since
that the probability of the indirect
ir (-8Í-)) -tq!) c^+rG*.
Level-Dependent Algorithms
4.7.L The Level-Dependent Logarithmic Reduction
Algo-
rithm
The level-dependent logarithmic reduction algorithm (LDLRA) was developed by
Bright and Taylor [3] and Ramaswami and Taylor [28]. For m > 2¿define the matrix
t\ to be
Ha: pd@+2,) <1Qn-2,) k x(t@¡z,))lx(o) e L(m)1.
Physically, HH, gives the probability that beginning in
tually reach L(m+
2¿)
matrix Llf , for m )
21,
under the taboo that
it
L(^)
(4.7.1)
the process will even-
does not ever visit
L(*-
Z¿). The
is defined to be,
Lt!): pd@-2') <t(m+2,) a x(t@-2,))lx(o) e L(m)1.
(4.7.2)
CHAPTER
4. MATRIX ANALYTIC
METHODS: AI{
INTRODUCTION
LIl,canbe interpreted as being the probability that beginning in L(m) the
eventuaily visits
L(*-
2¿) before
it
ever visits
L(m+ZI).
The matrices,
90
process
H!)
and
LI!), form the basis for the LDLRA.
The factor,
nflr,f*r, +
rrflnfl_r,,
(4.7.s)
is interpreted as being the probability that the process will return to L(m) after
visiting either L(m+21) or L(m-2¿) under the taboo that it does not visit L(m+21+1)
or L(m
-
Theorem
2t+1). The algorithm to determine
9
G^
for m
)
7 is as follows [28],
The family of matrices defi,ned by equatr,on (/, 6 2) ,is gi,uen, for m
)
7
ba,
¿-1.
Gm,
Ë
l--o
where,
for
(.
>
r[i]-,*,,
n
i:0
Ll¿l
Tn- r+21
(4.7.4)
t
0
Hlol
Lyl
H[¿+r]
eQ?)-'qf),
eqf))-'Q?,
(4.7.5)
(4.7.6)
Ë
(ul't tlr*r, +
t
(Htr'r L#"n*r,
rlt
ul.!r,)* u)') utn\r,,
(4.7.7)
rflr,
(4.7.8)
k:0
oo
r!+tt
+ tlt n)'!r,)r rlt
lc:O
An elegant physically motivated proof can be found in
[2S].
This chapter has been primarily concerned with many of the fundamental
as-
pects of matrix analytic methods. The MAP and the QBD play a prominent role
in what is to follow; we use both to define the MBT. The MBT is an alternative
representation of the binary-branch point ctMMTBP. In particular, the MBT representation of the binary-branch point ctMMTBP gives to the process a structure
where correiations between the branch lifetimes and branch offspring distributions
arise naturally. Furthermore, this representation allows us to exploit a much richer
algorithmic basis from which to obtain some interesting measures that are of use in
CHAPTER
4. MATRIX ANALYTIC
METHODS: AIü
TNTRODUCTION
91
biology; an algorithmic basis that is quite under-developed in the branching process
literature.
Chapter
5
Markovian Binary Trees
5.1
Markovian Binarv Tree: Definition
In this section we define and construct the Markovian binary tree as a particular example of a level-dependent QBD, which was discussed in Chapter 4. A
Markovian binary tree (MBT) is a level-dependent QBD process with states
(¡rr(¿),
able
ór(t),...,öw1r¡(ú)) defined on [Jpo{{f}
l/(l)
t {1,... ,r}r}.
denotes the number of branches alive
X(t¡ :
The random vari_
at time ú. For k
:
1, . . . , ¡/(ú),
/¿(ú) gives the phase of the k-th branch at time ú. The phase process in each of the
branches evolves as the phase process of an n-phase MAP, as described in Section
4.3. If there arc k
branches alive then there are nk possible states of the phase
process for the entire MBT.
We define the Kronecker product and sum of two matrices.
dimension
axb
and a matrix Y is of dimension
cx
e) then
of the matrices Z ØY , is given by the ac x be matrix,
ZØY:
znY znY
znY
zztY
znY
zzzY
zotY zozY
92
zouY
If a matrix Z is of
the Kronecker product
CHAPTER
5. MARKOVIAN
B/¡úARY TREES
Z
The Kronecker sum of two matrices
and
93
Y which are both of dimension ¿ x ¿ is
given by
ZØY:ZØI+IØY,
where 1 is the
o,
x a identity matrix.
We shall state and explain the infinitesimal generator matrix for the process and
then discuss its qualitative behaviour. The transition rate matrix for the MBT is
0
0
0
At') A\') Af)
a
0
0
A!:) A?)
Af)
0
00
00
0
0
A?o
0
A?
0
A[')
Let 1(k) by the nk x nk identity matrix, with 1(0) : 1. The ntr x nk-| matrix Af,)
for
lç
)
1 is given by
k-7
Ay):D,tr¡ adø J@-r-i)
(b.1.1)
j:0
Equation (5.1.1) embodies the fact that only one branch can become extinct at any
moment of time. The nk x nk matrix
A\r)
:
Af)
for lc ) 1 satisfies the recursion
Afk-t¡ o
Do,
(b.1.2)
with Al0) : 0. Once again the nature of equation (5.1.2) indicates that there is no
interaction between any of the k phase processes that are currently evolving.
Finally
Lhe
nk x nk*r matrices .1f,) to, aII k > 1 are given by
k-1.
Af):Ðí,
aBxl J@-t-i), k>
r
(5.1.3)
j:0
where the n x n2 matrix
B
governs the observable transitions of the process. The
expression in equation (5.1.3) has k terms reflecting the fact that there are k actively
evolving branches and that any one of these branches can give rise to a new daughter
branch via the action of the matrix
B. The independence of lhe k evolving
branches
CHAPTER
5.
MARKOVIA¡ú BI¡\IARY TREES
94
is clearly seen by the fact that there is no interaction between the k copies of the B
matrix
The state space of the process can be partitioned into levels, just as in the QBDs
of Chapter 4. The space of states in level k e {0}
uZ+,
denoted by L(k), are those
states that consist of k actively evolving branches.
Example 5 A model with three non-absorb'ing phases, {I,2,9}
The naturai extension of the MAP is to force both branches at a branch point to
be in the same phase immediately after the branch point. The matrix B for such a
model has the form,
1) (7,2) (1,3) (2,r) (2,,2) (2,3)
(Dr)r, o o
o (D')', 00
2 (Dr)", o
o
o (Dr)r" 00
, (Dr)r, o
J
o
o (D')t, 00
(1,
1
(3,
1) (3,2)
o
o
o
(3,3)
(Dt)tt
(D')rt
(5.1.4)
(Dt)tt
The rows indicate the phase that the branch was in immediately before
it underwent
an observable transition and the columns give the birth phase of the daughter branch
(the left digit) and the phase that the parental branch (right digit) is in immediately
after the observable transition. Thus a branch in phase i immediately before the
branch point will generate a daughter and parent branch that are in an identical
phase immediately following the branch point. Hence only B¿,¡¡ for
j : I,2,2 are
non-zero and correspond to the elements (Dr)o¡. The general case where the parent
and daughter branches can be in any phase immediately after the branch point has
the transition structure given bv
(5.1.5)
The eiemenl B¿,¡¡, of the matrix B gives the rate at which the parent branch in phase
i
spawns a branch point (observable transition) such that the daughter branch is in
CHAPTER
phase 7
5.
MARKOVIA¡ú BI¡\IARY TREES
95
whilst the parental branch is in phase k immediately after the branch point.
\Mhen pictorially representing a branch point, the parental branch is drawn as the
right branch and the daughter branch is drawn
as the
left branch.
The MBT is a special case of the continuous-time Markovian multi-type branch-
ing process as we shall show in Section 5.2. The point of departure from the ctMMTBP is an issue of interpretation.
Because the branches of an
MBT are governed
by MAPs, hidden transitions do not correspond to tree nodes. Observable transitions that spalun daughter branches, on the other hand, do correspond to nodes. In
the cIMMTBP, the hidden transitions of the MBT are called singular transitions,
and result in the transformation of one particle type into a distinct particle type.
In the cIMMTBP such singular transitions do correspond to nodes. By interpreting
the dynamics of a branch using a MAP, the time intervals between branch points
(or nodes) need not be exponentially distributed. Furthermore, the interval between
branch points can influence the offspring distribution, in particular, see equations
(4.3.2),, (4.3.3) and (4.3.4)
Suppose
k
{
in Chapter
4,.
that at time ú the process is in a state with rn branches and let branch
m be in phase
r.
Suppose the current state of the process is,
(^, ú, ...)
b, r)
cr
1 ... k-7 k k+I
...,
d)
rn
where the number beneath each branch denotes the label of that branch. The
following transitions are then possible:
o A hidden transition to
causes
phase
j I
r,, occurs with rate (Do),¡. This transition
the state of the MBT to becorne
(*,
a)
1
o An observable transition that
parental branch is in phase
j
b,
i,
c)
d),
k-7
k
k+r
rn
spawns a daughter branch in phase z whilst the
immediately foliowing the transition, occurs with
CHAPTER
5,
MARKOVIA¡\I BI¡üARY TREES
96
rate 8,,¿¡. The new state of the MBT is
(m-l\, aj ...)
,i,
b,
j
c,
...,
1 ... k-7 k k+I k+2
d)
m+7
The tree is oriented such that the parental branch is the right branch and is
associated with the k
+ 1-th label and the daughter branch is the left branch
and is associated with the k-th label. The branches that were previously
labelled
k+7,...,ffi
have been re-labelled
to k-12,...,m+L
o Finally a catastrophe occurs on branch k at a rate dr. This
to
cease
to exist and the new state
causes branch k
is
(^ - 7, a) .. . , b., cj ...) d)
1 ... k-7 k
m-I
where the branches that were previously labelied
ld
+ 1,. . . )m have been rela-
belledtok-1,...,m-7.
5.1.1 An Alternative Representation of the States of the
Process
Using the above representation for the states of the process we find that L(m) is
populated by a total of
n^ states, since there
are rn branches and each branch can
be in any one of n possible phases. However, states of the process can be represented
in a more compact form, whereby at any time
ú,
the state of the process is given by
the number of branches that are in each phase. For example, if we have rn living
branches, then the states of the process are given bv,
(^r,*r,.
..
,mn),
where m¿ is the number of branches in phase ,i, and such that,
n
D^o:*
i:7
CHAPTER
5.
MARKOVIA¡\¡ BnüARy TR.EES
97
Using this representation then, the number of states with m living branches is
mln-7
TN
This is just the number of ways of placing rn like objects into n different cells, as
can be found in any textbook on combinatorics, for example see [29].
The number of states in this representation is ciearly less than
n-,
however under
this representation we lose the ability to distinguish between parent and daughter
branches; all branches are essentially treated alike, modulo their current phase and
thus we lose any concept of ancestry. The major portion of this thesis is concerned
with analysing tree topologies and this state space representation will not allow us
to identify topologies
as we have
lost all relationships between branches and merely
know the number of branches in each phase. As a result, in the remainder of this
thesis we use the original representation since this representation does allow us to
keep track of the history of each branch and hence of the topology of the tree.
5.2 An MBT is a special case of a ctMMTBP
5.2.L Definition
The MBT is a special case of the continuous-time Markovian multi-type branching
process where each and every branch point may have 0, 1, or 2 offspring. We now
write the MBT as a branching process. Recall from Section2.4 that the probability
i
generating functions of the offspring distribution for each particle
¡(z)(s)
:
t
p(i)(j)sil
are given by,
...sr;.
(b.2.1)
it' "'inev'l
The MBT is a special case of the cIMMTBP with offspring probability generating
functions that are quadratic. The generating functions are therefore,
/(¿)(s)
:&*
B¿,it"
_ä*,W,r*Ð, (do),
sjst
,
(5.2.2)
CHAPTER
5.
MARKOVIA¡\T BI¡úARY TREES
for all i e {7,2,.
..
,n}.
98
Suppose that the process consists of one particle of type
,i,
we can describe the qualitative behaviour of the process quite simply. The particle
of type
i will live for an exponentially distributed amount of time with mean 1l@o)oo
at which point the particle will either,
1. die without giving birth to any new particle and this occurs with probability
d,¿f (do)¿,
or
2. die and transform into a single particle of type j + i.and this
probability,
(Do)o¡ I @o)¿,
occurs with
or finally,
3. die and give rise to two new particles that have types
j
and k and this occurs
with probability, Bo,¡rl (do)na.
5.2.2 Regularity and the Mean Number of Branches
Since all the derivatives of
f@(s) are clearly finite, the process
does not explode
[2]. The matrix of the expected number of branches in the MBT case is, by Section
2.4.2
M(t):
exp(.At),
(5.2.3)
(do)oko¡,
(5.2.4)
where
A¿j
and
*0,
:
-
å(,t
õo¡)(Do)o¡.
(Bo,¡r
å
where ô,7 is the Kronecker delta , equal to one
equations (5.2.4) and (5.2.5)
it
lf i, : j
- 6¿j,
(5.2.5)
and zero otherwise. FYom
can be deduced that
A:
where C is an n2
+ B¡,ni)
Do
+
BC,
(5.2.6)
x n counting matrix, with
Coj,*:
I{i:
k} +
I{j - k}.
(5.2.7)
CHAPTER
5. MARKOVIA¡ü BI¡úARY
TREES
99
For a process with three non-absorbing phases, {L,2,3}, the C matrix has the form,
723
11
2
0
0
72
1
1
0
13
1
0
1
27
1
1
0
22
0
2
0
23
0
1
1
31
1
0
1
,)¿
0
1
1
33
0
0
2
(5.2.8)
Therefore the expected number of branches at time ú is given by
M(t):
exp[(D6 + BC)t]
(5.2. e)
From Chapter 2.4 we saw that the process was sub-critical, critical or super-critical
depending on the dominant eigenvalue of the matrix
o if
À¿
< 0 the process is sub-critical
o if
À¿
:
o if
À¿
> 0 the
A.
Thus
0 the process is critical, and finallv
process is super-critical.
5.2.3 Probability of Eventual Extinction
Recall that the minimal non-negative solution of
z(s)
:
g,
(5.2.10)
is the probability of ultimate extinction of the process, see equation (2.4.22). Now
if )¿ < 0 the process will become extinct almost surely, and if )¿ > 0 then q (
e
CHAPTER
5.
MARKOVIA¡ú BI¡úARY TREES
100
component-wise. For the MBT we have,
0 :
uQ.)@)
(d,o)o(f@(")
-
",)
: (r,),(#,*_är,W,r* t
n
\
d'o
k,j:t
+ D,@o)¡nsn+
|k--7 |
(5.2.11)
B¡,in
S,¡S¡a
(do)o
-
S¿
i
(5.2.r2)
O:d,fDes+B(s8s)
(5.2.13)
le:l
This can be re-written in matrix form
B¡,¡,¡
i:r
s¡,s
as,
The probability of eventual extinction is the minimal non-negative solution to equa-
tion (5.2.13), [2]. \Me shall return to equation (b.2.13) in chapter z. we next
show that the macroevolutionary models that were discussed in Chapter 3 can all
be subsumed by the MBT model.
5.3 MBTs and Simple Macroevolutionary
Models
We begin by discussing the simplest of the models, the constant rates birth-anddeath model (crBD).
5.3.1 Constant Rates Birth-and-Death Model
This model is characterised by the fact that there is only one particle type. A particle
will speciate with probability >,lQ+ p) or will become extinct with probability
p,lØ+ p). The probability generating function for the process is given by,
p(s)
: rh*h"'
To develop the MBT version of the crBD we associate )/(À
and
¡tlQtp)
with drl@o), and apply the
M
(b.3.1)
*
¡r) with Br,rrl(do),
mapping, since as before we only
CHAPTER
5.
MARKOVIA¡\I BI¡\IARY TREES
101
consider unstable branches in line with the analysis of 126, Appendix A]. Finally, we
write the MBT transition rates
(Do)tt :
_(À + p)
(5.3.2)
BtJt : )
d,1
:
(5.3.3)
(5.3.4)
l.L.
5.3.2 Proportional-to-Distinguishable
Arrangements Model
The PDA model in its original incarnation stated that each distinguishable arrange-
ment of the labels on leaf branches of a given size are equally likely. The PDA
model was shown in Chapter 3, Section 3.10 to correspond to the asymptotic sub-
critical constant rates birth-and-death model. If in the MBT model the rates
given by equations (5.3.2)-(5.3.4) and
if p >
are
then as ú --+ oo the distribution of
^,
probabilities over the extinct tree topologies is exactly given by the PDA model.
Here we mention a subtle point, in a subcritical branching process as ú
---+
oo, the
only tree topologies that have a non-zero probability are those that have only extinct
branches, application of the mapping
M'
does
not remove any non-extinct branches
as there are none. Pinelis [26] provided an alternative
definition of the PDA, called
the multi-rate-PDA (MR-PDA). In the subcritical domain of this model, the mapping Mq needs to be applied, and after all extinct branches are pruned, all finite
quasi-stable topologies of a given size are equally likely. To demonstrate the versa-
tility of the MBT we shall derive an MBT with an identical distribution to that of
the MR-PDA model as ú --+ oo.
Recall that in the MR-PDA setting, the probability of a particular random tree
of topology Ta is given by,
P(ro):
/
¿r \"-r
\r-;*r,
(r(1))",
(5 3 5)
CHAPTER
tf lfø
I
:
".
5.
MARKOVIA¡\I BI¡\IARY TREES
702
The MBT does not contain quasi-stabie phases, instead it has two types
of phase:
o phase
0 which represents extinct species, and
o phases 7 to n which represent live species
In order to transform the MR-PDA into the MBT domain we therefore
¡
map the quasi-stable phase of the MR to the absorbing phase of the MBT,
phase 0, and
o exclude the extinct state of the MR-PDA.
What the second condition implies is that we map the space of finite non-extinct
quasi-stable trees to the space of extinct MBT trees and discard all the MR extinct
trees.
In the MBT domain the rate at which a branch becomes absorbed is given
d4: ___!_
t - p(Ø)'
by,
(b.3.6)
and the rate at which a branch gives birth to an identical daughter branch is given
by,
Bt,tt
It
:
b(r
- p(Ø)).
(5.3.7)
can be shown that
(Do)r,
- -d4- Btrt: -(1 -ZUe(Ð),
by using the fact that p2(Ø) + 2e@)(r
-
p(ø))
+
(1
-
p(Ø))'
:
(5.3.8)
1 and
p(Ø):d+bp2(Ø),
which is equation (3.11.1) from Chapter
(5.3.8) is the fact that (do)r
3.
What is interesting about equation
: -(Do)tr < 1 . This tells us that by discarding extinct
trees in the MR sense we are essentially slowing down the clock of the process, since
these false events never occur in the MBT.
CHAPTER
5.
MARKOVIA¡ú BI¡\IARY TREES
By analyzing the process
tree has topology
T'
(-Do)-tB
write the probability that a random
as
p(T')
since
as ú --+ oo we can
103
:
(-Do)-'B
(n(T¡3,07)
ø p(zr6,,l))
,
(b.3.e)
is the probability that the node [0] will eventually become an inter-
nal node, and p(Tú,o1) and eQ¡$,] are the probabilities that the daughter branch
and the parent branch eventually evolve into trees of topology
tively. In this
case the process has only two phases, an
7¡fi,01
and
7¡fi,r1 respec-
extinct phase, corresponding
to quasi-stability from the MR-PDA model, and one unstable phase, so that equa-
tion (5.3.9) is a scalar, that can easily be shown to be equal to
p(7"
) :
:
å¡"r,rro(T¡g,o1)n6¡3,¡)
1
t -;Mb(l - e(Ð)eØr6^o)n(T¡3,)'
(5'3'10)
Now, prior to any true binary branch point, that is, a branch point that generates
two branches that do not eventually become extinct, there may be any number
of branch points in which one of the two branches eventually generates an extinct
subtree. This possibility is reflected by the fact that for each true branch point of
the topology we must pre-multiply by 1l Q
-
2bp(Ø)), which is interpreted as giving
the mean number of false branch points that occur before the occurence of the true
branch point. Now, in the MBT-PDA interpretation we do not allow for extinct
trees in the sense of the MR-PDA model. In the MBT-PDA model, extinct trees are
precisely those trees that correspond to the quasi-stable portion of MR-PDA trees.
Using induction on equation (5.3.10)
equal to
it can be shown that
s-l '
's-r '
(r(1))''
e@))"
(u(t
zue(Ø))
p(T'):
' (=-=\
\t
-
the above equation is
(5'3'11)
where p(1) is the probability that an individual branch eventually becomes absorbed
into phase 0 before a birth, and is given by
1n
/ :ì-0,:,
(do)r'
p(1)
¡\
q,,
(r -zue(Ð)(t-p(Ø))'
(b.8.12)
CHAPTER
5.
MARKOVIA¡\r BDüARY TREES
704
Combining equations (5.3.11) and (5.3.12) we obtain,
s
p(T')
q
1
t-
(t
p(Ø)
-
zueQ)
(5.3.13)
This equation is identical to equation (3.11.5) from Chapter 3 except for the factor
IIG - p@).
Recall that
in the MR-PDA model, âry
branches
that eventually
became extinct are pruned from the evolving tree in order to generate the correct
topology. As an alternative way of thinking about this, we apply the Mq mapping
from the space of realizalions to the space of topologies. On the other hand, to
correctly obtain the measure for each topology, we cannot disregard the existence
of extinct subtrees; they must be taken into account. For the MR-PDA model, the
set of trees as ú ---+ oo is populated by two subsets of non-zero measure, the space of
topologies representing extinct trees and the space of topologies representing quasi-
stabie trees. The space of extinct trees, in the MR sense, are absent in the MBT,
and so to obtain the correct distribution we must divide the MR-PDA probabilities
by
1
- p(Ø) to obtain equation (5.3.13).
5.3.3 The super-PDA
model
The multi-rate interpretation of the sPDA (MR-sPDA) model was discussed in Sec-
tion 3.11.2. The probability that a random tree is generated with topology Ta
is
dependent on the number of unitary splits of that topology. The MR-sPDA allows
daughter branches to be born in the quasi-stable state, whereas the MR-PDA does
not. The MR-sPDA model can also be re-written in terms of an MBT. The space
of topologies that we are concerned with in the MBT version of the sPDA model
is precisely that space which consists of only the quasi-stabie topologies of the MRsPDA model. This is just the generalization of our analysis for the MBT version of
the MR-PDA in the previous section.
To transform the MR-sPDA to the MBT domain,
1. we wish to map the quasi-stable phase to the absorbing phase of the MBT
CHAPTER
5. MARKOVIA¡ú BI¡\TARY TREES
105
The problem with doing this directly is that in the MR-sPDA model, a branch
can give birth to a daughter in the quasi-stable phase, whereas, in the MBT we
do not allow births directly into the absorbing phase. Consequently, we map
the quasi-stable phase to a holding phase. Once a branch enters the holding
phase
it will then
be absorbed with probability one.
2. we exclude the extinct MR-sPDA state
The second point implies that at each transition we exclude the possibility that
a
branch and hence subtree will become extinct (in the multi-rate sense). Thus we
condition on the space of trees where extinction does not occur, that is, the topologies
with a positive number of quasi-stable branches as ¿ -+ oo; see the discussion in
Section 5.3.2. We map the space of these trees to the space of sub-critical MBT
trees whose branches have all been absorbed in phase 0.
MR-sPDA
Intermediate
q
r--n@
q
hp(Ø)
hpQ)
hQ -
MBT-sPDA
T-p@
p@))
zb(I-t)pT)\-p(Ø))
b(l- ?)(1 - p(Ø))'
b1
2b(I
b(l
-'r p
1-
1
l-1
-p
7-2b 7-1
- t)p(Ø)
-r)(t-p(Ø))
b(1
-ry)(t-p(Ø))
1-2b(1 -t)p@)
Table 5.3.1: Branch point transitions from MR-sPDA to MBT-sPDA
Table 5.3.1 has three columns. The first column tabulates the most important
branch point transition probabilities of the MR-sPDA model:
1. the rate at which a branch transitions into the quasi-stable state,
CHAPTE,R
5.
MARKOVIA¡\I BI¡úARY TREES
106
2. Ihe rate at which a branch point occurs such that the daughter branch is born
quasi-stable and the parent branch eventually becomes extinct ís hp(Ø),
3. the rate at which a branch point occurs such that the daughter is born quasistable and the parent branch does not eventually become extinct is ó7(1-p(Ø)),
4. the rate at which a branch point occurs such that either the daughter or the
parent eventually becomes extinct whilst the other does not is 2b(1
-l)pQ)G-
p(Ø)), and finally,
5. the rate at which a branch point occurs such that neither the daughter nor
the parent eventually become extinct is b(1
-
?)(1
-
p(Ø))'
As we have stated already, in the MBT environment we have elected to
ex-
clude any transitions that lead to extinction in the multi-rate sense. Extinction
in the MBT environment
coincides
with quasi-stability in the MR environment.
Consequently, we cannot allow any transitions in the MBT environment that could
potentially lead to tree extinction in the MR sense. Thus, following the explanation
given in Section 5.3.2, we divide all these rates by
t-p(Ø)
which then gives us the
second column in the table.
Consider now the transition 2b(7
l)p@) from the second column. In the MR
world, this leads to a branch point such that one of the branches generates a subtree
-
that eventually becomes extinct whereas the second one does not. In the MBT
world such a branch point, or such a transition does not exist. Why? Because
we
have chosen to not allow extinction in the MR sense. We divide all the rates in the
second column
by 1 - 2b(l
- l)pp)
because as we shall see this plays the role of
one of the diagonal elements of De. This factor is less than one and similarly to the
PDA example of the previous section
it
acts to set the "clock" of the process.
Because we are dividing the rates of the second column by what amounts to one
of the diagonal elements of Ds, the third column actually gives us the transition
probabilities for the MBT model of the process. We explain each transition in turn.
CHAPTER
5.
MARKOVIA¡ü BI¡úARY TREES
t07
But before we do, note that, the absorbing phase for the MBT is 0, phase 1 is the
holding phase and phase 2 is the unstable phase. Now, there are three ways in which
a branch can become extinct in the MBT sense, that is, enter phase
0. The frrst
probability
The
secondisgivenavffi,whoseinterpretationisquitesimple,itisthe
is via a direct transition from phase 2 with
probability that a branch in phase 2 will eventually undergo a transition to phase 0,
without any associated branch point. Of course, in the MR-sPDA environment this
was associated with an internal branch point, such that the parental branch became
extinct and the daughter branch directly became quasi-stable. This internal branch
point does not exist in the MBT,
by r_rfèif¡¿rb1, which
it
is a leaf node instead. The third way is given
gives the probability
that a branch, which was in
phase
2 immediately before the branch point, will undergo a branch point such that the
daughter will be born in the holding phase, phase 1. This daughter branch will then
become extinct with probability
final entry in the third colum",
1. The parental branch remains in phase 2. The
gives us the probability
ffiffi
that a branch
point occurs when the parent branch is in phase 2 such that the parental branch
remains in phase 2 immediately after the branch point and the daughter branch is
also spawned into phase
2. Given the
above transition probabilities we can easily
write the rate matrices for the process:
Do:
-1
0
B-
0
- (r
000
0 tu 0
(5.3.14)
-
2b(1
-
ùp(Ø))
0
(5.3.15)
b(1
-7)(1
-p(Ø))
and finally,
d-
1
q+btP@)
(5.3.16)
t-p(Ø)
We elect to study the sub-critical process as ú --+ oo, so a random tree will
eventually become extinct with probability one. Recall from Chapter 3 that any
CHAPTER
5. MARKOVIAN BI¡{ARY TREES
108
topology, and in particular, any extinct topology can be written
T' : {([r(o)],
so the probability
just given
[0])',7¡6,01 ,T¡3,r1],
that a random tree will be mapped to a tree of topology T'
is
by,
p(T"):
where
as,
(-Do)-'B
(-Do)-'a(n(r¡6,01)
ør(z¡6,,1)),
(b.3.17)
is the probability that the root branch will eventually undergo a
branch point, and p(Tó,o1) and eQ¡$,r) are the probabilities that the daughter and
parental subtrees will eventually have topologies
T¡6,o1
and T¡3,t1. Since each event
is independent we multiply them together, and the Kronecker product reflects the
fact that we need to keep track of the independent evolution of the two branches
emanating from the first branch point.
More explicitly then,
pr(Tr\,oìp'Q¡g,rt)
pt(T')
1
0
Pr(T')
0
t-zb(t-t)pQ)
1
It
000
0h 0
0
b(1
nr(T¡3,0)nzQ¡fi,a)
-7)(1 -p(Ø))
rr(T¡6,o)nt(T¡6,t1)
nz(T¡ï,o)nz(T¡fi,a)
(5.3.18)
Any tree that commences in phase 1 can only ever reach a
size of
one, th:us p1(T')
:
0
lf lT"l > Z. As a result only pz(T') is non-zero for lT"l > 2, thus
lfrP\
pz(T")
:
1
r_
(fup{T¡3,01¡er@¡3,)
2b(r _.y),p(Ø)
+ b(l
- 7)(1 - e(D)e"(T¡3,o)n"(T¡3,r))
(5.3.1e)
The flrst term of equation (5.3.19) is non-zero only if
since
it
commences
in phase 1, and in this
7¡fi,0¡
case pr(Tl6,o1)
is a single branch topology
:
?, (lTó,oll
: t) :
1. we
therefore re-write the above equation to reflect this,
Pr(T')
7l
r
-
ó(1
2b(r
- t)p@) vhI{lró,'::
:
7)p2(Tó,1)+
- ?)(1 - e(Ð)e"Øó,0ìnr(T¡6,rì)
(5.3.20)
.
CHAPTER
5. MARKOVIAN
BI¡üARY TREES
109
Re-arranging equation (5.3.20) leaves
b(l -?)(1 -p(Ø))
pz(T') :
'l
1-2b(I-t)p(Ø)
(t-ry)(t-p(ø)
r{lrfi,ql:
t)pz(T¡ï,¡)
¡pr(T¡g,o)pr(rfi,r)
(5.3.21)
The first term in the brackets of equation (5.3.21) can be re-written
'l
(r
-
ry)
(t -
p(ø))pr(lró,ol
{lrfi,oll
)r
because of the indicator function t{lf¡6,q1
as
: t}pz(T¡3,o)0"(T¡3,,),
: t}.
(5'3'22)
Combining equations (S.a.Zt) ana
(5.3.22) we obtain,
''l
pz(T') :
- ry) (r - e@)e"(lr6,otl :
xp2(T¡fi,o)P2(T¡fi,r1).
(r
r)
r{lrfi,ql :1}+1)
(5.3.23)
We can now extract the indicator function in equation (5.3.23) and place
it
as the
exponent of the expression in brackets,
pz(T")
:
b(l
- ?)(1 - p(Ø)) (
1-zb(t-t)p@) \tt - ?)(i -
r{l13,orl:1}
',
e@))ez(lz,6r1
: ¡ -,-'
xp2(T¡fi,o)pr6¡g,rt)
(5.3.24)
T" has size lT'l : with u unitary branch points,
"
then using an inductive argument, equation (53.2\ can be shown to be equivalent
Suppose the tree of topology
to,
s-1
.Y
nz(T"¡:
(r
- ry) (t - p(ø))p,(lró,otl :
r)
+1
(r,(t))"
(5.3.25)
Clearly,
and incorporating this into equation (5.3.25) the final result is,
s
+1
(5.3.26)
CHAPTER
where lS
tlQ -
:
5. MARKOVIA¡ú BI¡\TARY
TREES
110
This is precisely equation (3.11.17), except for the factor of
-ffi"6.
e@)). Again, for an explanation,
see
the text immediately following equation
(5.3.13) in Section 5.3.2.
5.4
The MBT and the Multi-Rate Model
Simple evolutionary models such as the constant-rates BD, the PDA and the sPDA
can be shown to be special cases of the MR model. AII three of these models are
characterised,
in the MR context, by the fact that they have only one unstable
phase, one quasi-stable phase and one extinct phase. These models are scalar (as
there is only one phase per class) and can be analysed. However, in order to provide
for better macroevolutionary models clearly more complex MRs are required, since
these simple models do not yield imbalances that are consistent with observation.
Non-scalar MR models have the major drawback that they cannot be conveniently
analysed because they do not have a representation that allows such an analysis to
be performed, see Chapter
3.
Even if such a representation existed, the MR model has one other complication,
the distinction between quasi-stability and extinction makes any transient analysis
of the process very difficult to perform. Branches that become extinct must
be
pruned from the tree. The consequences of pruning are such that the death of only
a few branches may have profound effects on the entire topology of the tree. Figure
5.4.1 depicts a tree that evolves under a MR model and at two different times we
perform the necessary pruning of the extinct branches to ascertain the tree topology.
Tree,,
Tf
To, is the complete tree with unstable, extinct and quasi-stable branches, tree
is the unstable topology of the tree after a pruning is performed at time
ú1
and tree T,! is the unstable topology of the tree after pruning is performed at t2.
Topologies
T"
and
Tf
are very different; pruning the
extinct branches has the effect
of producing longer branches, since any transition that generates an extinct branch
is not treated as a branch point after pruning. Thus the rate of undergoing a true
CHAPTER
5. MARKOVIAN
BI¡úARY TREES
111
-..,1 Extinct
+
time
IJnstable
lz
I
r1
TO
7",
Figure 5.4.7: An example of the pruning required for MR trees.
branch point, that is, a branch point that yields two non-extinct branches, is clearly
less than the rate of undergoing
just a branching event. Topologies
Tf
and
Tf
are
also very noticeably different; the death of three branches between ú1 and ú2, has
changed a four branch topology from the second topologically isomorphic class of
size four trees to a single branch topology. In other words, after pruning all extinct
branches at' t2 the tree has not undergone even one true branch point!
This highiights the critical problem that the MR faces by pruning extinct branches:
in order to prune correctly at any time, a complete historical knowledge of the tree
is required because of the profound effects of even a small number of extinctions.
Flom a mathematical perspective then, to avoid these issues one studies the process
in the limit
as
t
---+
æ and then prunes the extinct
branches.
Now, because the MR lacks a representation from which a usefui analysis can be
performed in the limit as ¿ ---+ oor we shall express the model in two different ways
that render it more transparent to analysis. The MR model can be transformed
CHAPTER
5.
MARKOVIA¡\T BI¡\|ARY TREES
t12
into an MBT-like representation or directly into a MBT representation. We shall
discuss both methods and then show that the MBT representation of the MR is the
preferred option.
5.4.L The MBT Representation of the MR Model
As a first step we shall directly transform the MR model into a standard MBT
representation, called the MBT-MR model. The impetus behind such an approach
stems from the success with which the MR-PDA and the MR-sPDA models were
transformed to MBT processes. Because the procedure we use in this section is a
generalisation of the methods employed in Sections 5.3.2 and 5.3.3, the MBT-MR
model reduces to those models when the phase space consists of one quasi-stable
and one unstable phase.
The model is correctly formulated by
o discarding ali the extinction
phases of the MR model as they are not counted
in the topologies, and
o mapping all the quasi-stable
phases
to a holding phase, say 1. Flom this phase
absorption subsequently ensues with probability one.
As a result, the space we are concerned with here is the space of MR trees that are
finite with all branches being quasi-stable. Once again, since extinction in the MR
sense is
not allowed in the MBT model, to transform the rates from the MR model
to the MBT domain we must therefore divide each rate by the appropriate
see Sections 5.3.2 and 5.3.3.
7
-
pi(Ø),
This implies that we are interested in the distribution
of the space of random trees from the MR domain whose quasi-stable portions are
mapped to Tq, and not the measure of the subset of trees that correspond to extinct
trees in the MR sense.
As before, leb U be the space of unstable phases, and Q be the space of quasi-
CHAPTER
5.
MARKOVIA¡\T BI¡\IARY TR.EES
stable phases. The absorption rate from any phase
Dora lhq
oo:ffi
'
113
i e U is given by,
|_Dqea.o¿qP¿(Ø)
(5'4'1)
This equation tells us that if a branch is in phase i e U then it will become absorbed
if it
undergoes
1. a direct transformation from i, to any q € Q, or,
2. abratch point occurs
such that the daughter is spawned in some phase, q
e
Q,
whilst the parent branch, still in phase i immediately after the branch point,
subsequently generates a subtree that becomes extinct in the MR sense.
We sum over all Q since we are not interested in which quasi-stable phase
but merely that it enters one. Thus the absorption vector d is given
it
enters
by,
(5.4.2)
where dy is the absorption vector for the unstable phases and is of dimension N (U) x
1, and where the first element represents the fact that a branch in phase 1 will be
absorbed with probability one.
The rate at which branch points occur from'i e U to a daughter branch in some
phase u
€U
is given by,
B¿,u¿
_ oiu(t -
p"(Ø))
7
(t -
pn(Ø))
(5.4.3)
- p{Ø)
A parent branch in phase z gives birth to a daughter branch in phase u with raïe oiu
and since we wish that both branches independently never become extinct,
multiplied by the probability of non-extinction which is (1 - p"(Ø)) (t
rate at which a parent branch in phase
i
-
o¿u is
po(Ø)). The
e U will give birth to a daughter branch
in phase 1, is given by
Bi
qo¿q(r
1
r
-
-
p,(Ø))
pi(Ø)
(5.4.4)
CHAPTER
5.
MARKOVIA¡ú BI¡üARY TREES
114
The total tate at which a parent branch gives birth to a quasi-stable daughter branch
occurs with rate Ðqçqotq. We require that the parent never becomes extinct (in
the MR sense) and to ensure this happens we multiply ¡V (1
-
p,(Ø))
To illustrate the transition structure, consider an example where there are two
unstable states denoted by
U : {2,3} in the MR, which
and 3 in the MBT, and two MR quasi-stable states
1 in the MBT. In this instance the matrix
1
ozt
2
t
T2
13 2t
0
0
I
0
t)
ozs
oM
I
ozs
2
Q: {4,b} mapped to the phase
B is of the form.
?2
00
0 o"r(I - pr(Ø))
00
0
are mapped to phases
23
31
0
00
0
0 oy(I -
osz(t
-
pz(Ø)
00
32
,)ù
0
p'(Ø))
0
oss(t
where for space reasons we have excluded the 11 column, which has only zero entries.
Let (Do)i,, foyi,u €l'1, i
phase e to phase
I
u, be the rate at which a hidden transition occurs from
u. This is given
(Do)i,
by,
¡ !i\-!"@)n,Q)
- --!vt-p¿\Ø)+ffi
(5'4'5)
This equation also has a similar interpretation: a branch in phase ,i can undergo
hidden transition into phase z either directly via
¡,t¿,
a
or indirectly via giving birth to
a daughter branch in phase z, whilst the parent branch eventually becomes extinct
(in the MR sense); once the parent is pruned all that remains is a single branch in
phase
u. The diagonal elements of the matrix Ds
(Do)oo:
-
d'o+D@o)¿,
u€U
for i'
eU
and (Do)tt
- -1.
are given by,
r t
ue{I, U}
B¿,u¿
(5.4.6)
The total rate at which a branch may undergo a birth is
given by the sum of all the possible eventual outcomes of a birth. These outcomes
are,
o both subtrees become extinct, or
-
ps(Ø)
CHAPTER
5. MARKOVIAN
BI¡\IARY TREES
115
o the daughter branch is quasi-stable and the parental subtree does not become
extinct, or
o the daughter branch is quasi-stable and the parental subtree does
become
extinct, or
o the daughter branch eventually becomes extinct
and the parental subtree does
not,
o the daughter branch does not become extinct while the parental subtree does,
or finally,
o neither subtree
Thus,
it
becomes extinct.
is relatively straight-forward to show that the rate of giving birth, is given
by,
:
Dø*D*'
q€Q
ueU
Doo,p,(Ø)po@)
u€U
+t
ois(r-pn@))
qeQ
+t
o¿qp¿(Ø)
seQ
tu€U (oo,p,(Ø) (t - po@)) t o¿u(t - p"(Ø))po@))
+ I oi,(L - p"(Ø)) (r - po(Ø))
+
(5.4.7)
utu
Now using equation (5.4.7)
(Do)*:
it
can be shown that (5.4.6) is equal to,
- (t -
(,.nr,o,u'u(Ø) *r,,,0,(r)))
It is also easy to see that if there is only one unstable phase,
(5.4.8)
then equation (5.4.8)
reduces to equation (5.3.14).
The probability that a random tree eventually has a topology of
p(T"): (-Do)-'a (n(T¡3,0,) ø r(z¡5,,1))
T'
is given by
.
(b.4.e)
Equation (5.4.9) can be solved analytically using a simple recursion on the subtree
topologies. Let
T"
be a topology of size s, then the recursion to solve for
T'
p(T") : Gnù-'a if lT"l: r
p(T") : (-Do)-'B (nØ¡|,o) s eQ¡|,r)) otherwise,
is
(b.4.10)
(5.4.11)
CHAPTER
5.
where lzr;,ql
It
MARKOVIA¡ü BI¡\IARY TREES
116
: j and lró,ul : s - i.
can be seen that the direct transformation from the MR to the standard
MBT model generates rather complicated transition rates between phases. The
alternative is to transform the MR into an MBT-like model. Such a transformation
is performed in the next section where
it will be seen that
complicated transition
rates are small price to pay compared to the difficulty in solving the equations that
give the probability of any topology, even for the one branch topology!
5.4.2 The MBT-like
Representation of the MR model
In the MBT-like representation
we
1. group all the quasi-stable states into phase
-1.
This phase is effectively a sec-
ond absorbing phase, the rate of absorption into this phase from any unstable
phase
i eU
is Dqee Fr,q, and
2. group all the extinct states into the second absorbing phase, 0. The rate of
absorption into this phase from 'i e U is given by
Ða.,
l.r¿¿. Ãny branch
that
eventually enters phase 0 is pruned from the tree.
Thus the reason why we call this an MBT-like structure is due to the presence of
two absorbing phases whereas an MBT is defined with only one absorbing
phase.
We shall call this model the MBT-like-MR (MBTI-MR) model. Furthermore, the
-1 phase is not treated as a traditional absorbing phase, since a daughter may be
spawned in phase -1. Let N(U) be the number of states inl,l. The transformed
model has the following hidden transition structure,
-1 0u
-1
0
0
0
00
00
u
a
dDs
(5.4.12)
5.
CHAPTER
where
a¿
:
MARKOVIA¡\T BI¡úARY TREES
Ðqee
ltr,q,
dt
:
Ðaeo
lL,¿¿
717
matrix of internal (hidden)
and where Ds is the
transitions between the states that are
inU.
The non-diagonal elements of Ds are, (Ds)¿¡
The dimension of D6is N(U) x N(U).
:
lh¡ il
iI j
and the diagonal elements
are given by,
(Do)on:
- Dron+
qeQ
t
tdeDþ¡¿r-D,po,+ ue{-I,u}
u€U
(5.4.13)
ou,i
At this point we would like to point out that from an analytical
it
would be much easier to just use the
-1
perspective,
phase like a holding phase, as we did
in the MBT version of the MR-sPDA model. However, because we wish to keep a
distinction between multi-rate quasi-stability and extinction we cannot do this. We
shall get around this by introducin g rhe (N (U) +
that,
r:f
where
OÚ
phase,
t) x
(N (u)
*
1) matrix, D, such
o o'.],
(5.4.14)
Lo nol
is the transpose of the zero vector 0. The first row now corresponds to the
-1.
Now any branch that enters phase -1 will never exit
it.
Let,
TI
:Lol'
a:l0
(5415)
|
be the vector that gives the rates of extinction from each phase. Notice how d-1 is
set to zero which distinguishes
-1
from phase 0, because a branch in
-1
remains
in this phase and it cannot become extinct, that is, enter phase 0. We shall also
see
that there is no communication between
-1
and 0 via the matrix B, the matrix
governing observable transitions. Consider) now the observable transitions. The
observable transitions are those transitions that generate a distinct daughter branch.
The transitions where a parent branch, in phase i, spawns a daughter branch in phase
i
Ç
{-7,L/} whilst the parentai branch
and o¿¡
=Ðq.qo¿o iÎ
j - -1.
remains in phase
i, occur at rate o¿¡ if j e fi
Therefore,
B¿,j¿:o¿j,j€{-7,U),
(5.4.16)
CHAPTER
5.
MARKOVIA¡\I BINARY TREES
118
B¿,¡¡,:0 for k + i,. Furthermore, B-1,¡¡:0 for aIIi,,j e {_7,U}. So, the
matrix B is of dimension (N(U) + 1) x (N(U) + 1)'. This process is conservative,
and
De*Beld,:O
(5.4.17)
The internal dynamics of each branch is governed by the matrix D6 just as in
the ordinary MBT. The matrix De is invertible so an analysis as ú ---+ oo is possible,
however the matrix D is not invertible. We introduce the matrix D, which is defined
by,
lo oú I
':Lo (D,) 'l
(541s)
The purpose of D is to keep the dimensions of the matrices in the expressions to
follow from becoming too skewed.
The space of extinct trees consists of those trees whose branches have all been
absorbed in phase
0. Let
commencing in phase
¿
po(Ø), for all
i e {-I,t/},be
the probability that a tree
eventually becomes extinct. The equation for p(Ø) is,
p(Ø)
:
(-D)-'à + eD)-r BeQ) ø e@).
(b.4.1e)
A tree may become extinct by direct absorption into 0 from the root branch, with
probability given by the first term of equation (5.4.19) or by first undergoing a birth
and then having both subtrees subsequently becoming extinct independently and
this is given by the second term in equation (5.4.19). Note that the probability
that a tree becomes extinct given that its root branch began in phase
is clearly equal to 0, since the phases
in Chapter 7 that p(Ø)
-r,
p_íØ),
-1 and 0 never communicate. It will be seen
can be found algorithmically.
Recall, from Chapter 3 that the topology of the quasi-stable portion of a tree was
denoted by To, in this chapter, we continue with nomenclature here. Let p¿(Tq),
,e{
- l,U},
be the probability that a random tree commencing life in phase z will
eventually attain a topology of Tq. Note that since we are studying the distribution
as ú
-+ oo and we are assuming that the process is subcritical, the only trees that
CHAPTER
5.
MARKOVIA¡ú BI¡úARY TREES
119
have a non-zero probability of occurring are those that have a finite number of
extinct and quasi-stable branches and zero unstable branches. Now, there exists
only one topology with lTø I : 1, and so we often write, p(1), for the probability
that a random tree evolves to eventually attain a one branch (quasi-stable) topology.
We can write the expression for this probability, and
:
p(1)
it
is given by,
(-D)-'ã,+ (-D)-L B(eQ) øeQ)) + (-p)-'s(p(t) sp(Ø)) t €_t, (5.4.20)
where
and e-1 is a vector of zeros, except for the
-1
position which is one. The inter-
pretation for equation þ.a.20) is straightforward. A tree can attain a one branch
quasi-stable topology in two ways) the first is directly via a phase transition to
and this has probability
(-D)-1â,
-1,
and the second is via a birth such that either the
daughter or the parent subtrees eventually become extinct and the other eventually
attains a one branch topology, and this has probability (-D)-tB(p(Ø) Sp(1)) +
?n)-tA (p(t) ø p(Ø)). Now if the tree commences in phase -1, it is with probability 1 that the tree will be of size 1, hence the presence of e-1. More generally,
n-r(Tø)
: I{lfo I : t} since any branch commencing in phase -1 cannot give birth.
The equation for the probability that a random tree will attain a topology of Tq
> z is given by,
for lTø
tt- |
p(Tn)
:
(-D)-'B
(n6¡E,q) ø r(2,f,,,,)) +
+eD)-l8
@Øn) sp(Ø)) +
(-D-18
þ(Ø) ø pgn)).(5.4.21)
This equation illustrates the fact that a tree with eventual topology Tq achieves this
by giving birth
1. to two non-extinct subtrees of topologies
{
(
[r(o)],
[01 ¡
r';r, Tó,ot,
Tt\,t
], o',
7¡å,01
and T¡3,t7 such
that 7q :
CHAPTER
5. MARKOVIA¡\I BI¡úARY
TREES
720
2. such that the parent or daughter subtree becomes extinct, whilst the other
evolves into a tree of topology Tq.
One must be very careful in expanding equation þ.a.27). Recall in Chapter 3, the
equation for the probability that a random tree eventually has topology Tq in the
MR-sPDA model is given by,
p(rn)
:
b(1
-
-,¡r@¡3,0)nØ¡3,a) + 2bp(Tq)p(Ø) +
hevó.,r)1{
113,q
|
: r}. (5.4.22)
Notice that the last term in this equation gives the probability that the daughter
branch at the first branch point is born quasi-stable and hence can only have a one
branch topology, so if Tq has a daughter subtree that has a size greater than 2, this
term is zero. Now, since equation (5.4.21) is the matrix generalization of the above
equation, when we expand
it there will be a term from (-D)-rBþ(Tö,r)
which will be non-zero only if the daughter subtree is of size
ør(z¡3,r1)
1.
The effects of disregarding extinct branches when mapping to the topological
domain, can be seen from equations (5.4.20) and (5.4.21). The last two terms from
both equations represent the scenario that, following a birth, either the daughter or
the parental subtree becomes extinct, and is subsequently pruned, whilst the other
subtree evolves to a topology of Tq.
The last two terms in equation (5.4.20) cannot be gathered because the Kronecker
product is non-commutative and so an analysis such as that performed for the
simple sPDA model cannot be applied, see [26] and Chapter 3. The equations
for the dynamics of this process are difficult to solve analytically even in the single
branch topology case because of the complicating effects of pruning extinct branches.
Algorithms can be developed to solve equations (5.4.20) and (5.4.2I)
The probiems associated with pruning were not an issue in the standard MBT
representation for the MR model since all the extinct states from the MR model
were discarded and quasi-stability was treated as extinction. The distribution of
the standard MBT model is identical to the distributiorr, p(Tq) of the MR model
CHAPTER
5. MARKOVIA¡ú BI¡{ARY TREES
721
restricted to the space of random trees whose quasi-stable portions are mapped to
1t.
To illustrate the simplicity in using the MBT-MT model, consider the three
branch topology depicted in Figure 5.4.2. The probability of attaining this topology
e
T [0,0I---
[0,1]
Figure 5.4.2: A three branch topology
is given by,
p(T'): (-Do)-'a
where
lfó,rfl:
1 and
,
(5.4.22)
lTó¿l: 2. Since the daughter topology
consists of a single
(n(rfi,01)ør(z¡6,,1))
branch we have that
pT¡ï,o): (-Do)-Id,
Ø.4.24)
and since the topology of the parental subtree is a two branch topology, using
equations (5.4.11) and (5.4.10) we obtain,
e@¡|,a): (-Do)-18 [(-r0)
-1d, Ø
(-¡o)-'d]
(
5.4.25)
Substituting the expressions for p(T¡$,01) and p(T¡$,r), into equation (b.4.23)
we
finally obtain,
p(r"):
(-Do)-' a (f- n,l-L d, Ø (- Do)-, Bl{-ro)-', s (-D0)-'d]
)
Ø.4.26)
This example demonstrates the power of transforming the MR into the MBT-MR
model. The three branch topology has an analytical solution whereas the single
branch topology in the MBTL-MR model requires algorithmic methods to solve!
CHAPTER
5.
MARKOVIA¡\T BI¡úARY TREES
t22
The implicit nature of equations (5.4.20) and (5.4.2I) represents the interaction
between the full topology and extinct subtrees are absent in the MBT-MR equations;
this is the result of neglecting the extinct states in the MBT-MR model. This
difference is most striking because equation (5.4.11) can be solved using recursion
and matrix multiplication for all s
)
2,, whereas
the most simple equation in the
MBTL-MR model, equation (5.4.20), is implicit and requires an algorithmic
scheme
to solve! Finally, the asymptotic distribution of the much more simple MBT-MR
model is still identical to that of the MR model.
The MR-PDA, and MR-sPDA models consider only topologies that consist en-
tirely of a finite number of quasi-stable branches, almost surely. The crBD model
considers topologies
that consist entirely of unstable branches, and the crBD in-
terpretation of the PDA model (crBD-PDA) considers topologies that consist of
a
finite number of extinct branches, almost surely. The MBT representation of all the
above models considers topologies that consist of a finite number of extinct branches,
almost surely, except for the crBD model which considers unstable topologies. In order to be completely unambiguous about the types of topologies we are considering
when having discussed each model, we have been meticulous in denoting the type of
the topology of a tree as T" and the type of a topologically isomorphic class of size
s as IFf,", where
z:
a,u,Q,e
.
FYom
this point and throughout the remainder of the
thesis we shall no longer make such a designation as ail models will be discussed in
terms of the MBT in which case the types of the topologies that are being considered
are unambiguous.
The first flve chapters have seen us discuss the need to provide useful models
of the macroevolutionary process. The most simple models have been shown to be
deficient in a number of areas, particularly when one factors in the need to generate
topologies with imbalances that mimic those of phylogenetic trees. The most com-
plex model to date, has been the MR model of Pinelis [26]. In theory
it
provides a
significant amount of flexibility that can account for the variable imbalances found
in nature. However, this model suffers from some quite fundamental problems. The
CHAPTER
5.
MARI<OVIAN BI¡úARY TREES
123
MR model of Pinelis, which is a special case of a ctMMTBP, cannot be analysed
in a practical fashion. There is a well
developed theoretical approach
to the ct-
MMTBP, however very little of this theory provides any practical use in a modelling
context. Furthermore, the MR model requires one to prune extinct branches from
the topology, an extremely difficult task to achieve in any transient analysis, and
still of considerable difficulty in the limit
as
t
---+
oo. Thus, in order,
1. to provide a physically reasonable, flexible model of macroevolution, and
2. to provide a model that is amenable to practical algorithmic analysis,
we have given an alternative representation of the binary-branch point cIMMTBP
in the language of QBDs, known as the MBT. In doing this, we have given to the
binary-branch point cIMMTBP, which is a structure that has very little correlation
between particle lifetime and offspring distribution, a very rich correlation structure.
The correlations are caused by considering the evolution of each branch in terms of
a MAP.
We have thus far shown that the MBT model subsumes all the simple macroevo-
lutionary models. F\rrthermore, the flexibility of the MBT is such that we
have
provided two alternative representations for the MR model. The first representation
is in terms of an MBT-like model and the second representation is in terms of the
standard MBT model.
Our next step is to begin demonstrating the power of the algorithmic approach
and the flexibility of the MBT as a model. In Chapter 6 we provide an algorithm
that calculates the distribution of imbalance conditioned on tree size. Using this
distribution we than calculate the mean of Colless' index of imbalance. We
show
that there exists a one parameter family of MBT models that can generate mean
imbalances that have a range spanning from a lower mean imbalance than the crBD
model all the way to the theoretical maximal mean imbalance. The correlations that
abound in the MBT domain, due to the MAP, produce interesting effects that can
be exploited in developing a suitable macroevolutionary model.
Chapter
6
Probability Distribution of
Imbalance
6.1
Introduction
Chapter 3 provided an introduction to the types of models that have been utilised
in current phylogenetic research. As stated in that chapter, the imbalance of phylogenetic trees is playing an increasingly important
role. The
reason is
that the
imbalance can be used to infer information on how the rates of speciation or ex-
tinction have changed over time, because higher or lesser imbalances are likely to
have been generated by variations in these rates. The simplest macroevolutionary
models predict imbalances that are either too low, for example the constant rates
BD (crBD) model 170,22,301, or too high, for example the PDA modell22,26,30l.
For a graphical representation of this point for varying tree size, see Figure 5 from
Rogers [30]. The reason why neither of these simple models can predict the correct
imbalances is that they do not have the capacity to allow the rates to vary across
a tree; the rates of speciation and extinction are fixed. As a result, the ability to
vary rates over time and throughout a topology gives one the propensity to generate
topologies with the imbalances that are more closely aligned with those found in
r24
CHAPTER
6. PROBABILITY DISTRIBU?/O¡ú OF IMBALANCE
725
nature. More recently, there has been a greater emphasis on developing models that
allow for rate variation. The most sophisticated model to date, that proposed by
Pinelis [26], is a special case of the ctMMTBP.
The MBT model, as developed in Chapter 5, is also based on the cIMMTBP.
However, a novel interpretation and representation allows us
to develop
complex
interactions between the phases of each branch, and as a result, allows for significant
rate variation in different parts of the tree and at different times. Furthermore, the
MBT model subsumes all of the other important macroevolutionary models that
have been discussed here, including the multi-rate model of Pinelis.
The purpose of this chapter is to demonstrate further this flexibitity by developing an algorithm that can calculate the mean imbalance for the MBT when we
condition on the size of the trees we are interested
in. This follows the work of
Rogers [30] who developed algorithms to determine the moments of the imbalance
for the constant rates BD and PDA models, when conditioned on tree size. We will
further show that we can flnd a very simple one-parameter family of MBTs that has
the flexibility to produce the entire range of theoretically possible mean imbalances.
In Section 6.2 we discuss the algorithm for determining the mean imbalance of
an MBT model given its size. In Section 6.3 we discuss the mean imbalance for
the simple models, crBD, PDA, s-PDA and Completely Unbalanced (a special
case
of the sPDA model), and then using the algorithm from Section 6.2we show that
there exists a one-parameter family of MBTs that has the flexibility to generate any
mean imbalance. Finally, in Section 6.4 we discuss the computational complexity of
the algorithm.
6.2 The Imbalance Algorithm
In this section, we wish to deveiop an algorithm that can calculate the mean imbalance, using Colless' index of imbalance, for the MBT model. To do this, we must
first calculate the distribution of imbalance, conditioned on tree size.
CHAPTER
6.
PROBABILTTY DISTRIBUTIO¡\T OF IMBALANCE
726
We know that for any given size s, Colless' index of imbalance must lie in the
set, {0, 1,..., (llZ)(s
have imbalance
'd
-
1)(r
- 2)).
Let C¿,, denote the set of trees of size s that
for all ? € {0, 1,..., (tlZ)(s
1)("
-
- 2)}.
Using the sets
C¿,s we
partition the space of trees of size s, '1f" into
1
2
s- 1)( s- 2)
?TF
trs
U
-
\1.,
i:o
"
If p[C¿,"] represents the probability of the space C¿,", in other words the probability that a random tree has a topology that is in
then the mean of Colless'
C¿,",
index of imbalance is given by,
iplco,']
: /2(s-t)(s-2)
ti:o D;-1";'rr"-Ð p[c¡,,]
t
E[1"1s]
(6.2.r)
since each C¿," is disjoint. Consequently, to determine the mean of Colless' index of
imbalance we need to first determine the probabilities, p[C¿,"].
Recall, that we can represent a tree of topology
T
as,
f : {(.(0)1, [0])(c) ,T¡o,o1,T¡0,r1],
where 7[o,o] urd Tlo,rl ur" the topologies of the daughter and parental subtrees, respectively. The imbalance of a tree of shape
T
can be determined from the shapes
of its constituent daughter and parental subtrees, T¡o,o1and
r"(T)
Let
It
IF¿,"
:
r"(T¡0,o1)
+ r"(T¡o,tì +
7¡6,11,
in other words,
lrl - zlr¡o,qll
be the ú-th topologically isomorphic class of size s, where
t
9.2.2)
:
1,2,.
..
,Tr.
is clear, then, that any topology from this class has the same imbalance, because
rotating the subtrees at each internal node does not change equation (6.2.2) since
llTl
-
2lT¡o,o1ll
also easy
: llrl -
to see that
2lT¡o,r1ll
si'ce
lTro,ql
:
lTl
-
lTto,t
l. F\rrthermore, it is
each imbalance class, C¿," is the union of the topologically
isomorphic classes of size s that have an imbalance of
t that have the property,
ci,,:
{tll"¡n',,"1
: z},
i.
Let C¿,"be the set of indices
CHAPTER
6. PROBABILITY DISTRIBU?IO¡\I OF IMBALANCE
I27
where I"[F.r,"] is the imbalance of one member, and hence all the members, of the
topologically isomorphic class IF¿,". We then have that,
ci,,: [J u,,".
t€C¿,s
\Me can generate
the set of trees in a non-empty class,
bining trees from the non-empty classes
C¿,7
C¿,s,
r€cürsively by com-
and Cn,"-j at node [0] provided that,
i,: ll k+
ls -2jlwith Co,r consisting of only the single branch topology. As we
shall see the algorithm to determine the mean imbalance given tree size is based on
this recursion. The set
C¿,"
:
C¿," can be
{{(tr(o)1,
written
[01¡r';1,
Ct,j,Cn,"-j]li:t+k+1"
U {ttlrto)1,0)(o),
i|C4,j
f
as,
-zjl]
ct ,"-i,c,,i}li:t t k+ls _ 2il},
(6.2.3)
C¡a,,-¡ and
C¿,"
:
{{(tr(o)1,
[01¡t';1,
Ct,j,Ct
,"-j]lt: t + k + lt - zjl]
9.2.4)
otherwise. Figure 6.2.1 depicts all the non-empty imbalance classes for trees of size
1 through
5. Commencing from C6,1, we see that
C6,2
is constructed from two
C6,1.
Proceeding in this manner one finds that Cs,3 is empty and Ct,s is non-empty since
the combination
Co,r
with
C6,2 gives
us an imbalance of
I. :
0
+0+
13
-
2l
:
t.
Similarly, C3,a is constructed by combining Cs,1 and C1,3 and so on.
Using equations (6.2.3) and (6.2.4) the probability that a random tree has
a
topology from C¿,", p[C¿,,], is given by
p[C¿,"]
: (-Dù-LB t
(n[auÀøp[Cr,"-r] +I{C.t,j
I
Cn,,-i}p[C*,"-r]øp[a¿,r]).
{t,k,i}€s.j
(6 2.5)
Suppose we wish
to determine the probability distribution of imbalance
classes for
trees of size s. We use the following recursive procedure:
Set
p[Co,']
since
it
: (-Dù-|d,
consists of only one branch. Then, loop through all tree sizes, 2
1t 1 s.
CHAPTER
6. PROBABILITY
I=0
DISTRIBUTIO¡ú OF IMBALANCE
I=l
I=2
I=3
ø
C
C,,,
C
Size
I=4
I=5
728
I=6
co,t
co,,
2
C
3
,3
C
4
0,4
5
3,+
C
5,J
65
Figure 6.2.1: An illustration of the imbalance algorithm
o
For each tree síze,t,loop through all the imbalances, 0 <
i <712(t-I)(t-2).
:0.
-
Set p[C,,r]
-
Loop through all possible daughter subtree sizes, 1 < I < t
x Loop through all possible daughter subtree
-
imbalances,
7.
0(
l¿
(
tl2(t-1)(t-2).
. Test to
see whether there exists
a non-empty imbalance
class
Cro¡4 such that,
ri:'i-l¿-lt-ztl,
with 0 1r¿
1112(t-l-1)(t-I-2)
and if
C¿0,¿
is also non-empty
then set
pl0¿,r)
:
plC¿,rl+ (-Do)-t B(nlC,t,,i8p[C"',¿_¿]
I {C,u,t
I
*
C,u,t_-t}plC,n,t-ù ø p[C¿,,¿]).
In Section 6.3 we begin by computing the mean imbalance for the simple macroevoIutionary models we discussed in Chapter 3, for size 5 trees. We conclude that section
CHAPTER
6. PROBABILITY DISTRIBUTION
OF IMBALANCE
729
be demonstrating the flexibility of the MBT by developing a one parameter family
of MBTs that can generate any desired mean imbalance, conditional on size five
trees. This model also generates some interesting and surprising features for higher
size trees.
6.3
Some Results
for Simple Models
In Section 6.2 lhe mean of Colless' index of imbalance, equation (6.2.1), was expressed
in terms of the imbalance classes. In most of the previous work on deter-
mining the mean imbalance, [30], a different probability measure was used. Rogers
[30] utilised a measure based on partitioning trees into their topologically isomor-
phic classes and not in terms of the imbalance classes. However, since the imbalance
classes consist of the union of topologically isomorphic classes there are fewer imbal-
ance classes, and therefore our imbalance algorithm has to calculate fewer probabil-
ities. Nevertheless, we shall determine the mean imbalance for some simple models
using the topologically isomorphic class approach which for these simple models is
straightforward, particularly for small trees.
As before let IFr," be the ú-th topologically isomorphic class of size s, where
,7"}, and ?l is the number of possible topologically isomorphic
trees of size s. Let 1"(lFr,") be the imbalance of this class. Recall that,
ú
€ {1,
...
E"[1"]
:
classes
for
lE[I"1t],
the expected value of Colless' index of imbalance conditional on size s trees. This
can also be written
as
,,r.s
E"[1"]
:t1"(ß',,")p"[]Fr,"l.
(6.3.1)
t:L
In Sections 6.3.1-6.3.4 we show how to find the mean of Colless' index of imbalance
for some of the simplest macroevolutionary models. We then show in Section 6.3.5,
using the imbalance algorithm of Section 6.2, that the MBT has sufficient flexibility
6. PROBABILITY DISTRIBUTTO¡\I
CHAPTER
OF IMBALANCE
130
to cover all possible mean imbalances for a given tree size. To do this we generate
a simple four phase one-parameter family of MBTs.
6.3.1 The Constant Rates BD Model
The space of trees, IFr,", can be generated by joining, IF,,, and F*,"_r, where
{1,... ,T¡}
j
and k e {1,
...,7,_¡}.Note, for
each IFr," there oniy exists one
I
€
l, k and
such that,
rk" ¿'s
- [ t([r(o)],
[01¡t';l
t
Harding
classes
18]
,ß,,i,Fr,"-i] u {(["(0)], [01;rrt,F*,"-¡,lr,,r] if rn¿,j rFn,"-¡
if IF¿,j : F*,"-,
{(["(o)], [01;tzl, F,,i,F*,"-i]
first derived the probability distribution of topologically isomorphic
for the constant rates birth model. The probability of a random tree of
size
s attaining a topology from IFr,", is given by
}'niltr,l2
P"[lFr,"] :
finilv
1r
i:
sl2
k k:
t
(6.3.2)
r,)n"-, [F¿,"-¡] otherwise
where IF*,, and F,,"-, are the topologically isomorphic classes of the two subtrees at
node [0]. We also showed these equations to be valid for the constant-rates birthand-death model in Chapter
3.
Example 6 Mean'imbalance for
It
sr,ze
5 trees
is easy to deduce that there are three topologically isomorphic classes for size five
trees, since size five trees are constructed from,
o size 1 and
o
size 4 trees, or
size 2 and size 3 trees.
There is only one class for each of the size two and size three trees, and there are
two classes of size four; one consists of joining two size two trees and the other one
consists of joining a size three tree with a size 1 tree. As a result, this implies that
CHAPTER
6. PROBABILITY DISTRIBUTION OF IMBALANCE
Classl
131
Class 2
t-2
I=3
Class 3
I=6
Figure 6.3.1: The three topologically isomorphic classes of size 5
there are three classes for size 5 trees. A member from each class is depicted in
Figure 6.3.1.
It
is straightforward using equations (6.3.2) to show that,
2
Pultrr,r]
Ps[tr2,s]
trs[lFr,u1
6
:
1
6
3
6
:
2,
, I"(Fr,u) :
3,
, 1"(1F3,5) :
6.
, I"(Fr,r)
The mean imbalance, equation (6.3.1), is then,
Es[I"]
:2 xã21325
*, " 6 *u * ä :
-ã
:
4.16667
CHAPTER
6. PROBABILITY
DISTRIBUTIO¡\I OF IMBALANCE
732
6.3.2 The PDA Model
The PDA model states that each topology of a given size is equally likety. The PDA
can be represented as a constant rates
or as an MR model
birth and death model in the limit
or as an MBT,
as ¿ ---+ 69,
5. Slowinski [33] first derived
the equations for the imbalance distribution for the PDA model. In this section,
1261,
see Chapter
we derive a much simpler set of equations that generate the distribution. Pinelis
[26] showed that there is a correspondence between the number of distinguishable
arrangements and the number of topologies given the tree size. Due
respondence we derive a set of equations
to this
cor-
that give the PDA distribution on the
topologically isomorphic classes. Recall from Chapter 3 that the number of distinct
topologies of a given size can be found using the following recursion equatiorr,
s- I
^t
with boundary condition, ¡fi
:
Ii:1 r4&-n,
(6 3.3)
1. Therefore) we can see, for example, that there is
1 topoiogy of size 2, 2 topologies of size 3, 5 topologies of size 4 and 14 topologies
of size
5.
Recall from Chapter 3, that the number of topologies
in a topologicaliy iso-
morphic class is given by the number of uneven branch points. Let the number of
uneven branch points of the topologies in IFr," be denoted by e¿,", then the number
of topologies in lFr,", l/(lFr,") is
N(Fr,")
: )et's
(6.3.4)
All the topoiogies of a given size are equally likely, so the probability that a tree of
size s comes from IFr,", is given by,
P"
Example 7 Mean,imbalance for
ltr, ,"1
"l
sr,ze
:
ff(F''"
¡/"
5 trees.
)
'
(6.3.5)
CHAPTER
6. PROBABILITY DISTRIBUTION
Figure 6.3.1 indicates that, N(Fr,")
8. As a result we
:22:4,
OF IMBALANCE
¡/(1F2,")
:27:2
and N(Fr,")
133
:23:
have,
4
Prltrr,¡]
14
2
Prltrr,¡]
Pr[Fs,r]
:-
2
(6.3.6)
7
:-
1
74
7
8
4
74
7
(6.3.7)
(6.3.8)
So the mean of Colless' index of imbalance is therefore given by,
rc5[1"1s
- b] :, "?*
3x
T
* u" ]
:l
: +.+zesr
(6.3.e)
6.3.3 The sPDA Model
The PDA model discussed in Section 6.3.2, states that all topologies of a given size
are equally likely. The sPDA model on the other hand, states that the larger the
number of unitary branch points a topology has, the more likely
it
is to occur. A
unitary branch point is a branch point where lhe daughter subtree consists of only
a single branch. Recall from Chapter 5 that for the MBT version of the sPDA the
probability of obtaining a topology of size s with u unitary splits given that the
process began in phase 2, was
pz(T)
1
t-
p(Ø)
s
os-7
+1
lr
(6.3.10)
where,
a-:
rJ
b(t-t)
T- %e _;)ee).
(6.3.11)
In this section we are interested in understanding the distribution given the size of
a tree and as a result, for trees of size s, the factors, p"-1,
s
CHAPTER
and
1
S\ZE
S
6. PROBABILITY
- p(Ø) are not important
Tlees
DISTRIBUTIO¡\I OF IMBALANCE
134
because they are identical for all those topologies of
in a topologically isomorphic
class do not all have the same number of
unitary branch points, because rotating the internal nodes of a topology has the
effect of swapping parent and daughter branches, and thus changing the number of
unitary branch points. As stated previously, the topologically isomorphic class,
is represented
]F ts
IF¿,",
as
{ (t'(o)1,
[01 ¡
rrr,F
r,,i,F,,"_,
{ (t"(o)1,
] U { ( tr(o)J,
[01;
rrr,F
¡01
r,,i,F,,"_,
¡r,r, F,,"_¡, F*,j
]
} if
if
I
:
IF,-,
tuil
IFk,j
ß,,"_¡,
IF,¿rò-../
^ ..
(6.3.12)
Now since the position of a single branch subtree determines whether the branch
point to which
it
is attached is a unitary branch point or not, we introduce two
measures, No(Fr,", u) which gives the number of topologies in IFr," that have u unitary
branch points, if the trees in IFr," are joined as the parental subtree at some branch
point, and ÄI¿(lFr,,, u) which gives the number of topologies in
lFr,"
that have u unitary
branch points, if the trees in IFr," are joined as the daughter subtree at some branch
point. The number of topologies in
ìFr,"
that have u unitary branch points when
considered as a parental subtree is given by
r) : É
No(Fr,",
U:L
(nOtr- ,,,a)Nr(ß,,,-j,u
-
u) +
1{F¿,"-¡ *Fr,}No(F,,"-r, a)Nr(Fr,,,"
where, No(Fr,r,0)
and I
:
1 and,n/o(F., J,u)
€ {1,2,...,7,_¡}.When
:
0, for u
t
- r)),
7, and where k e
(6.3.13)
{1,2,...,T¡}
considered as a daughter subtree,
obeys
^¡d(iF¿,,,u)
the same recursion, but has a clifferent set of boundary conditions, given by l/¿(F'r,r, 1)
1, and No(trr,r, u)
:
0 for u
f
1. We need the different boundary conditions because
single branch daughter subtrees give us one unitary branch point, whereas a single
branch parental subtree does not contribute any unitary branch points. The sum-
mation in equation (6.3.13) commences from
u:
7 because all daughter subtrees
have at least one unitary branch point, and finishes at ø
:
u since considered as a
:
CHAPTER
6. PROBABILITY
DISTRIBUTTO¡ú OF IMBALANCE
135
parental subtree a single branch has no unitary branch points.
We now give an example of calculating ÀI¿(1Fr,", u) and No(Fr,",
u). There are two
topologically isomorphic classes of size four, lFr,n and ìFr,n. We shall apply equation
(6.3.13) to JFr,r. We first note that, this class is constructed from size one trees and
size three trees. Now, there is only one topologically isomorphic class of size three,
lFr,, and
it
is obvious that there are only two trees in F-r,r. One of the trees in ß-r,,
has only one unitary branch point and the other has two, thus
Nr(Fr,r, 1) :
1) : 1, and
No(Fr,r,2) : ¡/o(F'\s,2) : 1.
(6.3.14)
^fr(Fr,e,
(6.3.1b)
Using the above we can calculate, for example, Ne(E2,4,2),
2
Ne(F24'2)
:iu*':'ï,-i'i; ;;] li'* "î'.
"'2 - i))
l/¿(Fr,r, 1)¡/e(tF1,1, 1) + ¡/d(F 1,3,2)¡/e(ß'1,1,0).
Consider the first term, l/¿(Fr,r, l)¡/e(lF'r,3,1). Now ÀI¿(trr,r,1)
single branch forms the daughter subtree and ,nr/r(F'r,r, 1)
N¿(tr,,r, l)No(trr,., 1)
: t x 1:
:
:
1 because the
1, so,
1.
The second term, ÀL(trr,r,2)No(trr,r,0), is zero because, l/¿(lFr,, ,u)
:
The third term, ÄL(Fr,.,1)Nr(Fr,r, 1), is also zero because l/r(F'r,r, 1)
single branch parent subtree does not generate a unitary branch
0
:
if u I
0, since
7.
a
point. The final
: t because ÀL(Fr,r, 2) : t from equation (6.3.15) and
: t. As a result, ¡/(1F2,4, 2) :2.
term, l/¿(trr,s,2)Nr(Fr,r,0)
No(Ft,r,0)
Thus equation (6.3.10) tells us that the probabiiity of a random tree of size
s
having a topology that is an element of IFr,", in the sPDA model is
P"[Fr,"] :
D:i
tÊ,
No(F,,","¡
(r + ojm)"
t;:1¡ro(p,,", "¡ (r +
cjm
(6.3.16)
where p(1) is the probability of a size one tree and is given by equation (3.11.16)
CHAPTER
6. PROBABILITY DISTRIBU?IO¡ü OF IMBALANCE
136
Example 8 Mean'imbalance for size 5 trees.
Our first step is to calculate the numbers of trees with u unitary branch points for
each of the topologically isomorphic classes. In order to avoid
repititious calculations
that require equation (6.3.13) we shali give the results for the non-zero
values:
ffo(trr,r,2) :
2,
(6.3.17)
:
ffo(Fr,r,2) :
ffo(Fr,r,3) :
No(Fr,r,1) :
Nr(Fr,r,2) :
Nr(trr,r,3) :
No(trr,u,4) :
2.,
1,
1,
1,
(6.3.13)
b:
Ø:
No(Fr,r,3)
We choose, for example, the values,
0.3,
(6.3.19)
(6.3.20)
(6.3.21)
3,
3,
1.
(6.J.22)
(6.3.23)
(6.3.24)
0.3,
d,:0.4
and
7:
0.5 for
the sPDA model. For these values of the parameters, the only solution to equation
(3.11.11) for p(Ø) from Chapter 3 is given by,
1- (t- 4b(r- r\d.\r/2
p(Ø):Ë:0.4247,
(6.3.25)
and thus,
,r\ :
P\r)
q+b"lp(Ø) ._.:o.4lTT
\r.1.,. ¡
1 _ rb(I _if@ -
(6.3.26)
r
and,
1
+
(1rtf6 : 3.3e41.
(6'3.27)
Using equations (6.3.17)-(6.3.27), the denominator of equation (6.3.16) can be shown
to be equal to, 439.8054. As a result, we have that,
Nr (Fr,u, 2) (3.3941)2
Prltrt,¡]
pr[]Fz,¡]
Ps [Fe,¡]
+
¡/e (F1,b, 3) (3. 3941 ) 3
439.8054
:
:
0.1151
0.6547
_ n,?n,
CHAPTER
6.
PROBABTLITY DISTRIBU?IO¡\T OF IMBALANCE
737
Thus, finally, the mean of Colless' index of imbalance for the sPDA model is given
by,
Eu[ä :2x0.2302*3 x
6.3.4 The Completely
0.1151
f
6x
0.6547:4.7339.
(6.3.28)
lJnbalanced Model
The completely unbalanced (CU) model is just the MR-sPDA or the MBT-sPDA
model with 7
: 1. The B matrix for the MBT-sPDA,
nience, is
B:l
[o o o
lo tu o
o
b(1
reproduced here for conve-
I
.1,
j
(6.3.29)
-?)(1 -e(Ø))
which becomes, in the case of the CU model,
0000
0b00
Bwhen
j : 7. Consequently,
(6.3.30)
all branches that are in phase 2, can only give birth
to daughters in phase 1, while the parents remain in phase 2 immediately after the
observable event. Since all branches
in phase 1 eventually become extinct almost
surely, all the daughter subtrees are single branches. Consequently, only topologies
with the maximum allowed imbalances are possible, that is, the completely
unbalanced topologies. Figure 6.3.2 depicts such a tree for size 5.
Example
I
Mean'imbalance.for si,ze 5 trees
Since the only possible topology in this case is the maximally unbalanced topology,
this implies that,
Et[I"]
:6'
(6'3'31)
CHAPTER
6.
PROBABTLITY DISTRIBUTION OF IMBALANCE
Figure 6.3.2: The completely unbalanced tree of size
138
5
6.3.5 A One Parameter Family of MBTs
In the preceeding sections we obtained the mean of Colless' index of imbalance for
the crBD, the PDA, the sPDA and the CU models for trees of size 5. We have
shown theoretically,
in Chapter 5, that the MBT
subsumes these models. This is
good, however is this good enough? We need a model that can not only account
for these simple models but has the ability to show more complex behaviour and
produce mean imbalances that mimic those found in phylogenetic trees. Indeed,
it
is the purpose of this section to show that a one-parameter family of four-phase
MBTs has sufficient flexibility to give us any mean from the entire range of possible
mean imbalances, as that parameter varies from 0 to 1. This demonstrates that the
MBT does have this required flexibility.
The more complex behaviour of the rates of speciation and extinction in the MBT
arise due to the interactions
that can be introduced between the underlying
phases.
We are therefore free to choose transition rate matrices for the process that show
very high correlations between the phases. Consequently then, the initial phase of
an MBT can have profound effects on its subsequent evolution. To demonstrate how
CHAPTER
6. PROBABILITY
DISTRIBUTIO¡\T OF IMBALANCE
139
profound these effects can be, we study size 5 trees; at size 5, interesting behaviour
begins to emerge.
We commence by defining some terminology. A hidden transition from phase ø
to phase b is denoted by, ø --+ b, and an observable transition in which a branch
is in phase ø immediately before the branch point, produces a daughter branch in
phase
i
immediately after the branch point, and a parental branch that is in phase
7 immediately after the branch point is denoted by a
---+ 'i,
j.
Low Imbalance Model
Consider a simple four-phase model that consists of phase 0, the absorbing phase,
and phases \,2,3, and which has transition rate matrices given by,
Do:
-10
0
0
0
-10
0
0
0.000001
-10
(6.3 32)
and
B-
00009.99999000
0
0000
0 0009.99999
0000
0 000
0
The above model has the following qualitative behaviour:
o a 1 ---+ 2,2
observable transition occurs with probability 0.999999,
o a 1 --+ 0 hidden transition occurs with probability
o a 2 ---+ 3, 3 observable transition
o a2
--+
0.000001,
occurs with probabilitv 0.999999,
0 hidden transition occurs with probability 0.000001,
o a 3 ---+ 2 hidden transition occurs with probability
o a 3 ---+ 0 hidden transition
0.000001,
occurs with probability 0.999999
(6.3.33)
CHAPTER
6. PROBABILITY DISTRIBUTION
OF IMBALANCE
740
1
0:
-cr(O):1
0,1
2+
0,r:3
0,1
0,1
2
2+
0,1,0:3
2
1,1:3,3
1,1:3,3
2
1,1,1;3,.
F
F
I .5
F
2.5
-) .5
Figure 6.3.3: Low imbalance MBT model
These transitions have been chosen specifically for the interesting tree shape dynam-
ics for trees of size 5 that ensue.
Figure 6.3.3 depicts a representative from each of the three topologically isomor-
phic classes of size five. The labels tþ : j, k mean that immediately after node [r/]
the daughter branch is in phase j and the parental branch is in phase k. The most
probable transitions that must occur in order to generate these topologies have also
been depicted. Note, that in this case, each and every topology within a topologi-
cally isomorphic class is generated by the exact same transitions as depicted in the
figure, just in a different order. The representative topoiogy from ìFr,u has only one
transition that is low probability,
o a hidden transition from phase
The representative topology from
3 to phase 2 along branch ([0, 1], [0, f , O1;ftl
ìF2." has
three low probability transitions and these
are,
o branch ([0], [0,0])(") 6""omes extinct from phase
2,
o a hidden transition from phase 3 to phase 2 along branch ([0,1],[0,1,0])(i),
and along branch ([0,1], [0, t,11¡tzl.
The representative topology from IFr,u also has three low probability transitions,
CHAPTER
r
6. PROBABILITY DISTRIBUTION
OF IMBALANCE
branch ([0], [0,0])(") b""omes extinct from phase
74r
2,
o a hidden transition from phase 3 to phase 2 along branch ([0,1],[0,1,1])(i),
and along branch ([0, 1, 1], [0, 1, 1, 1])(i).
Consequently, when considering size five trees, one would expect that p1[lFr,u] should
be much greater than either pr []Fz,¡] or p1[lPr,u], since there is only one low probability
transition in each of the topologies in
IFr,u and IFr,u.
IFr,u
and three low probability transitions in
When we apply our imbalance algorithm to this model we find that the
mean of Colless' index of imbalance for size flve trees when starting in phase 1 is very
near to /"[trt,s]
value is that
it
:
2 as we would expect. What is interesting about this particular
is considerably lower than that of the crBD model as calculated in
Section 6.3.1. That this is achievable, is testimony to the dependency that can be
generated amongst phases in an MBT. Thus, phase 1 is almost forced to generate
two phase 2 branches, phase 2 is almost forced to generate phase 3 branches and
phase 3 branches are essentially forced to be absorbed. Such dependency is absent
in the crBD, PDA and sPDA models.
High Imbalance Model
We now show that we can find an MBT that generates maximally imbalanced size
5 topologies. Consider the
MBT with
Do:
-10
0
0
0
-10
0
0
0.000001
-10
(6.3.34)
and
0 4.999995 0 4.999995
00
00
00
00
Its qualitative behaviour is such that,
o a 1 -) 3,1 observable transition occurs with probability
.4999995,
(6.3.35)
CHAPTER
6. PROBABILITY DISTRIBUTIO¡\I OF IMBALANCE
o a 1 ---+ 3,3 observable transition occurs with probability
o a 1 -+ 0 hidden transition occurs with probability
0.4999995,
0.000001,
o a 2 ---+ 1, 1 observable transition occurs with probability
0.999999,
o a2 ---+ 0 hidden transition
occurs with probability 0.000001,
o a 3 ---+ 2 hidden transition
occurs with probability 0.000001,
oa3+
r42
0 hidden transition occurs with probability 0.999999
o(0):1
cr(0):1
0:3,
0:3
a(0):1
0:
-
0,1:3,3
1..3,3
2-
0,0:l
0,0:1,
0,0,0:3,3
0,0,0:3
1,1:3,1
0,0,1:3,3
1,1,l:3,.
F
1.5
F
Fr.,
2.5
Figure 6.3.4: Maximally imbalanced MBT model
Figure 6.3.4 depicts the most probable representative topology from each of the
three size five topologically isomorphic classes of size 5. The transitions along each
branch have been included in the figure, and as before,
',þ
if [/]
is some node, then
,lc,l tells us that immediately followingllþ], the daughter branch
the parental branch is in phase l. The topology from
IFr.o has
is in phase k and
two low probability
transitions:
¡
there is a hidden transition from 3 ---+ 2 along branch ([0], [0,0])(¿) and
o branch ([0,0],[0,0, 1]) becomes extinct from phase 1, that is, a 1---+ 0 hidden
transition
CHAPTER
6. PROBABILITY
The representative topology from
DISTRIBU?IO¡\T OF IMBALANCE
IFr,u has one
743
low probability transition, the hidden
transition from 3 ---+ 2 along branch ([0], [0,0])(i).
The only topology from all the size five topologies that does not have any low
probability transitions is the completely unbalanced topology; the third in the figure.
This particular topology is by far the most likely topology because all the other size 5
topologies require at least one low probability transition. This includes all topologies
in IFr,, except for the one depicted in the figure. rffhen considering trees of size 5, we
would expect lhat p1[lF.,u] dominates because of the completely unbalanced topology
depicted in the figure. So the mean imbalance should be very close to /"[lFs,s]
:
6,
when starting in phase 1. In fact, when one applies the imbalance algorithm from
Section 6.2 this is indeed the case, E[1"15]
:
6.
A One-Parameter Family of MBTs
We are now in a position to discuss the one-parameter family of MBT models that
transform from our low imbalance model to our high imbalance model. The infinitesimal rate matrices for this model are,
Do:
-10
0
0
0
-10
0
0
0.000001
-10
(6.3.36)
and
0
B-
e.eeeee(
0
0 0 0 e.eeeee(l -c) 0 4.999995( 0 4.e99995c
000
0
0 0 0 e.e9999(1 -()
000
0
0
0
0
0
(6.3.37)
where
( e [0,1].
The graph in Figure 6.3.5 plots the relationship between the mean of Colless'
index of imbalance, conditioned on size 5 trees, and the value of
(
for trees com-
mencing in phases 1 and phases 2. Although, we do not analyse the transitions
CHAPTER
6. PROBABILITY
DISTRIBUTIO¡ü OF IMBALANCE
r44
Mean of Colless' lndex of lmbalance for Size 5 Trees
6
5.5
Phase
1------->
5
sPDA
PDA
crBD -------->
o 4
o
l-
(d
(õ
-o
3.5
.E
c(d
o
3
2.5
<-
Phase 2
2
Figure 6.3.5: Mean of Colless' index of imbalance for size 5 trees
that generate size 5 trees commencing from phase 2 we plot the mean imbalance
for trees commencing from this phase because
initial phase is. Also plotted
it
demonstrates just how crucial the
are the mean imbalances for the crBD,
PDA, and sPDA
models.
When
(:
0, the MBT reduces to the low imbalance MBT we discussed at the
beginning of this section. So, considering size 5 trees commencing from phase 1 we
find that the mean imbalance is very close to 2 as expected. At the other extreme
is a model very similar to the high imbalance MBT we discussed earlier. In this
case, the mean imbalance is very close
to 6. When
( is small but not zero we see
that there is a spike in the mean imbalance to three. Figure 6.3.6 magnifies this
region and depicts what happens as
(
becomes non-zero: as
(
---+
1
x 10-5 the mean
6. PROBABTLITY DISTRIBUTIO¡ü OF IMBALANCE
CHAPTER
745
imbalance approaches 3. In other words, this means that the probability of obtaining
topologies from C3,5 : Fz,s is approaching one. To explain why this occurs, we study
Mean lmbalance for SmallZeta
3
-
2.9
pþ¿ss'l
2
2
o
o
C
d2.6
(ú
-o
c
!2.s
c
(d
o
22.+
2.3
2.2
2.
2
0
o.4
0.6
Zela
0.8
1.2
x 10-5
Figure 6.3.6: Magnification of the mean imbalance for small (.
Figure 6.3.7, which once again depicts one tree from each of the three topologically
isomorphic classes of size 5 and the transitions that can generate these trees have
also been depicted. When
(
o the hidden transition,
is small there are four low probability transitions,
3
-- 2,
o the observable transition,
1
--
3,1,
o the observable transition, t -- 3,3,
and
o the observable transition, 2 ---+ I,1.
All topologies from
IFr,u have
4 low probability transitions, and these are,
CHAPTER
6. PROBABILITY
DISTRIBUTIO¡\T OF IMBALANCE
I
I46
cr(0):1
1
0:3,
.))
0,0:
L:3,1.
:3,I
0,1
2
0,1
,l:3,3
1.,1.,1.:3,1
F
2
1.5.e
F
J
F
2.5.e
6
3.5.e
Figure 6.3.7: One topology from each topologically isomorphic class of size
5.
o the observable transitions at [0], [0, 1], [0, 1,1] from t -> 3,1, and
o the observable transition at [0,1,1, 1] from
All the topologies from
JFr,.
1
--
3,3.
only have one low probability transition, a hidden tran-
3 ---+ 2. In the tree depicted in Figure 6.3.7 this occurs on branch
([0, 1], [0, 1,0])(¿), but it could occur on any of the other three leaf branches. The
sition from
topology from ìFr,u that is depicted also only has one low probability transition; the
observable transition
at [0] from 1 --+ 3, 1. The other topology from
IFr,u has two
more low probability transitions. We can therefore discard the topologies from lFr,u
as
occurring with extremely low probability and concentrate only on those from
IFr,u
and from IFr,r; in IFr,, the topology that we have depicted dominates. So, for low
( > 0, why does the topology
from ß'r,u and hence
ß-r,u
predominate over JFr,u?
The answer to this question resides in the relative sizes of the probabilities of
the 3
---+
2 followed by a
2 ---+ 3,3 transition
and the 1 --+ 3, 1 transition. The
probabilities of each of these transitions are given by their respective elements from
(- Do)irtBz,g¡:1X
x 10 '(1 - () and for t -- 3, 1 we have (- pù¡l81,3r :
the matrix product (_ Do)-tB. For 3 ---+2,2
10-e9.99999(1
- () :
9.99999
-
3,3, we have,
CHAPTER
6. PROBABILITY
(0.1)(4.999995C)
:
DISTRIBU?/O¡\r OF IMBALANCE
0.4999995(. Now the ratio of these two transitions is
(-Do)rr-Bt,' ,
50000000.ç_
(- Do)l] Br,r,
1 - ('
\Me can see from
the above equation that when (
probabilities are about the same, however,
if ( :
(-"'ìii+*
(-Do)li Br,"
Hence, for
(:
1
x
likely than the 3
probability of the
t47
10-5, the 1
---+
---+
3, 1
È
:
3.3
1
x
(6.3.38)
x 10-8 the orders of the two
10-5 we have that,
boo
(6 3 3e)
transition is roughly five hundred times more
2 --+ 3,3 transition and this is reflected in the fact that the
JFr,u
to 0.007936 for the 1Fr,u class;
class is 0.99206 as opposed
about 126 times more likely.
As
(
---+ 1,
00
B
--+
9.99999
0
00
0
0
0
0
0
0
00
00
00
4.999995
0
0
0 4.999995
00
00
which is identical to equation (6.3.35) and as a result, EtI"l5] --+ 6; as shown in the
graph of Figure 6.3.5. There are essentially three competing transitions from phase
1 as
(
gets larger and they are,
o the 1 ---+ 3, 1 which occurs with rate 4.999995(,
o the 1 -- 3,3 which occurs with rate 4.999995(,
o the I
-
2,2 which occurs with rate 9.99999(1
and
- ()
The first transition tends to favour less balanced topologies, the second transition
favours small size topologies and the last transition tends to favour more balanced
topologies. For small
( the third transition
gies result, however as
(
dominates and so more balanced topolo-
approaches one the first two transitions dominate, thus
generating small highly imbalanced topologies, until
of 6 is achieved.
(:
1 when a mean imbalance
CHAPTER
6. PROBABILITY DISTRIBU?/O¡ú OF IMBALANCE
148
The above one parameter family of MBTs was constructed in order to demonstrate the flexibility of the MBT. Specifically, for size 5 trees we constructed matrices
that generated the desired correlations amoungst the phases in order to show this
flexibility by achieving the complete range of mean imbalances. This model also delivers very interesting behaviour for larger size trees. Figure 6.3.8 graphs the mean
imbalance against the parameter
(
for size 6 trees. The minimum value that the
Mean of Colless' lndex of lmbalance for Size 6 Trees
10
I
Phase
o8
o
c
G
1
\
(E_
al
E
c(ú
g6
5
4
3
Figure 6.3.8: Mean lmbaiance for Size 6 Trees
mean imbalance is allowed to take for size 6 trees is 2, when lhe C2,6 imbalance class
predominates, see Figure 6.2.7. However, as can be seen from the graph when
the mean imbalance is approximately, 3.7. This implies that at
classes
that predominate
gies only require two 3
Cz,a.
0
0 the imbalance
are C2,6 and C5,6. They predominate because these topolo-
---+
2 hidden transitions, whereas all other topologies require
more. The mean is slightly above 3.5 because
lhan
(:
(:
Cs,a
is populated by more topologies
CHAPTER
6. PROBABILITY
DISTRIBUTIO¡\T OF IMBALANCE
749
The second interesting feature of the graph in Figure 6.3.8 is the spike which
occurs for small
(.
This spike occurs because as ( increases from zero the
C7,6 class
dominates. This imbalance class is generated from the iFr,, and JFr,, topologically
isomorphic classes, and for reasons similar to those given above when discussing size
5 trees, ìFr,u is the dominating topologically isomorphic class for small
(, hence, size
6 trees that are generated from IFr,u must therefore also predominate.
The final interesting feature of the graph is the gradual decrease in mean imbalance from near 7 to about 3.88 at around
( :0.61.
Since the mean imbalance dips
below 4, this implies that C25 is gradually acquiring a more prominent role. With
(
increasing, the dynamics amongst phases becomes more difficult to analyse, how-
ever, the number of low probability transitions that need to occur for topologies in
C2,6 are decreasing and as a result the likelihood of the occurrence of C2,6 topologies
increases and
that of
C7,6 topologies decreases.
in the probability of 2
---+
1, 1
and
t --
This is due, in part, to an increase
3,3 transitions; transitions that favour more
balanced topologies. Consequently, a number of the topologies in C25 can now be
generated without any low probability transitions.
All the previously studied models, other than the crBD,
have the common fea-
ture of predicting imbalances that are higher than those found in nature. In this
section we have given an example of a one-parameter family of MBTs that has the
flexibility of generating a very broad range of mean imbalances. This model predicts mean imbalances that encompass all the predictions of the models discussed
in this thesis, and much more. The MBT model is therefore extremely flexibie and
because of its numerical tractability provides an excellent model of the macroevo-
lutionary process. The final section in this chapter is devoted to determining the
computational complexity of the imbalance algorithm we developed in Section 6.2.
CHAPTER
6.4
6. PROBABILITY DISTRIBU?IO¡ú OF IMBALANCE
150
The Complexity of the Imbalance Algorithm
In Section 6.2 we discussed the details of the imbalance algorithm. To calculate
the probabilities of the imbalance classes of size s) we saw that we needed to first
determine the probabilities of all the imbalance classes of sizes
f < s. We reproduce
the algorithm here, for convenience,
Set
p[Co,']
since
it
: (-Do)-rd,
1t 1 s.
consists of only one branch. Then, loop through all tree sizes, 2
o For each tree
síze,
t, loop through all the imbalances, 0 < i < llz(t
-
1)(t
-2)
:0.
-
Set p[C¿,,]
-
Loop through all possible daughter subtree sizes, 1 < I < t
-
7.
x Loop through all possible daughter subtree imbalances, 0 { l¿ (
712(t-1)(t-2).
. Test to see whether there
Crn,r_'t such
exists a non-empty imbalance class
that,
r¿:,i-li-lt-zl,
with 0 1
r¿
1Il2(t-l-L)(t-l-2)
and if
C¿0,¿
is also non-empty
then set
plC¿,rl
:
plC4,rl+ (-Do)-t B(nlCt'ù8p[C"',,_¿]
I {C,0,,
I
C,o,t_-t}plC,n,r_.À
ø
p[C¿0,¿]
)
f
.
Let G(n,s) be complexity of the algorithm for size s trees with n-phase MAPs.
G(n, s) is then given by,
"
]1r-r¡1t-z¡
¡4
llt-r¡çt-z¡
G(n,s):I t t t
t:2 i:o
t:7
t¿:o
s@),
(6.4.1)
CHAPTER
6. PROBABILITY DISTRIBU?IO¡\I OF IMBALANCE
151
where g(n), gives the number of calculations required for the matrix multiplications
of the last step of the algorithm. This equation can be understood when one notices that the first summation results from the frrst loop over tree size, the second
summation is over all the possible imbalances of trees of size ú, the third summation
is over all the possible daughter subtree sizes and the final summation is over the
allowed imbalance values for the daughter tree. All other steps in the algorithm are
O(1), including the
if statement.
Now G(n, s) is an overestimation of the actual
complexity of the algorithm since, in actual fact, if a suitable triplet does not exist
to meet the conditions of the if statement then the matrix multiplications are not
performed.
The functionG(n,s) is tedious to calculate exactly, consequently, we shall instead
determine only its leading term. For large s, then, we have
G(n,s) x Ët"Ë-z)H
t:2
S
i:O
7/
2\
1) (¿- 2)
T D
i:o
t:t )'"n(,)
1" gl')
at"
1_
D
t:2 it"n(")
7 o,
n
72 r\ )
(6.4.2)
Thus the algorithm has complexity O(s6 g(n)), however as we have stated previously,
the actual complexity is less than this due to the fact that many suitable triplets
do not exist and so the matrix multiplications do not need to be performed. Figure
6.4.1 depicts the CPU running time for the imbalance algorithm for a three phase
model for tree sizes ranging from 1 to 50. Together with the imbalance algorithm
we have graphed the function
f(s)
:
7
72 "n''.
to demonstrate that, O(s6g(n)) is clearly an overestimation. We chose the factor
4.2 purely in order to demonstrate that O(s6g(n)) is greater than the actual com-
6. PROBABILITY
CHAPTER
DISTRIBU"IO¡\T OF IMBALANCE
t52
putational complexity of the algorithm.
x 10
2
s
CPU Time versus Tree Size for a Three Phase MBT
1.8
1.6
1
q)
E
12
tr
Ð
È
1
o
0.
I
0.6
0.4
0.2
0
0
Tree Size
30 35 40 45
50
Figure 6.4.I: The computational complexity of the imbalance algorithm
The MBT representation has allowed us to develop an algorithm that can calculate the distribution of imbalance given tree size and hence deduce the mean
imbalance. This algorithm is of polynomial order, with exponent lower than 6 and
probably quite significantly lower than
6.
The MBT model has two promising attributes that set
it
apart from the other
models thus far proposed. The MBT is
o extremely flexible,
as demonstrated in Section 6.3, and
o amenable to the use of algorithmic techniques.
In contrast, the MR modei of Pinelis [26] suffers from an awkward representation
that essentially closes the door to relatively efficient numerical analysis. What is also
CHAPTER
6. PROBABILITY
DTSTRIBUTION OF IMBALANCE
153
promising is that by representing the binary-branch point cIMMTBP as an MBT a
whole new world of possibilities in the domain of algorithmic analysis has opened up;
see
Chapter 7 for a further demonstration of the ability to develop algorithms for the
MBT, in this
case algorithms
that determine the probability of eventual extinction.
Chapter 8 presents a generalisation of these algorithms to the Markovian tree, the
matrix analytic representation of the general ctMMTBP.
Chapter
7
Algorithmic Approaches for the
MBT
7.L Introduction
As already discussed in previous chapters, the MBT is a special case of the ctMMTBP,
see
for example, Chapters 3 and 5. The theoretical basis of the cIMMTBP
is well established, [2, 27], however very little has been done in developing an algo-
rithmic basis that can be used to determine measures that are of
use
in a modelling
context, [5]. Dorman, Sinsheimer and Lange [5] provided a step towards rectifying
some of this problem by providing an algorithmic approach to a ctMMTBP with
Poissonian immigration. For example, they were able to calculate the probability of
extinction of the process at any time. However, as they acknowledged, their algorithms failed in the important supercritical
case
case as
time gets large. The supercritical
is the important case because the probability of eventual extinction is always
one for the sub-critical and critical cases.
In this chapter we provide two interesting algorithms that can determine the
probability of eventual extinction of the MBT process in the supercritical case,
and in Chapter 8 we generalise these algorithms to the general Markovian tree. The
754
CHAPTER
7, ALGORITHMIC
APPROACHES FOR THE MBT
155
theory of branching processes, [9], tells us that the probability of eventual extinction
is the minimal non-negative solution to equation (5.2.13) of Chapter 5, which
we
reproduce here,
o: d,f Dss +B(s8
Pre-multiplying equation (7.1.1) bV
(-ro)-1
s)
.
(7.1.1)
and re-arranging we obtain,
s: (-Do)-rd+ (-Do)-tB
(s
I
s)
(7.1.2)
Also note that the equation for the probability measure of the extinct space of trees
for the MR model, as expressed in equation (5.4.19),
p(Ø)
:
(- D)-'d + (- D)-r B þ(Ø) ø p(Ø))
,
(7.1.3)
has an identical structure to equation (7.1.2). Therefore, the algorithms developed
here also find application in determining p(Ø) for the general MR model.
The remainder of this chapter is organised as follows. We discuss tree labelling in
Section 7.2 followed by the Depth algorithm in Sections 7.3. The sample path classes
of the Neuts algorithm are transformed to binary tree topologies in Section
7.3.1
and as a consequence a neat description of the sample path classes of each iteration
of the Neuts algorithm is given in terms of binary tree topologies. The concept of
the order of an MBT is defined in Section 7.4 followed by the Order algorithm in
Section 7.5. Section 7.6 compares these two algorithms and shows that the Order
algorithm converges at a faster rate. The final section, Section 7.7, discusses the
logarithmic reduction algorithms and shows that, perhaps surprisingly, the Order
algorithm is still the most efficient.
7.2 An aside: Tree Labelling and Representation
In Chapter 3 we discussed a node labelling system which we restate here.
that we are at a node labelled
[rþ]: [0,rr,...,i,n],
where
Suppose
,it,...,i* e {0,1}, the
CHAPTER
7. ALGORITHMIC APPROACHES FOR THE MBT
function o takes us from node
[T/]
156
to node,
d(rþ)
:
ir,
[0,
That is, a(1þ), moves from the current node
..
.,ù^_t]
[T/]
up the tree to its parent node. Recall
that we define each node with respect to node [0], consequently, the root node is
Iabelled [t(O)]. As a matter of nomenclature, when we are referring to the function
o acting on a node [ú] *" do not
referring to the node [a(T/)] then
it in square brackets, however, if we are
we do encase it in square brackets.
encase
The portion of a branch that exists between the two nodes, t*(rþ)] and [r/], is
the ordered pair, (1"(ú)l ,lrþl).We write (t"(ú)] ,lrþl)(ù if this branch is an internal
branch. We write (1"þþ)1, [',/])(") if this branch is an extinct branch, and finally, we
write (ltþþ)],[rþ])(") if this branch is unevolved. If a superscipt is not specified then
we
just refer to that branch generically; its branch type is unimportant.
The function á takes us from a node [r/] and moves us along the parental branch
at
11þ]
to [rþ,1]. In other words for nodes
0(rþ)
and d(a(0))
:
lrþ]
I
l*Q)],
lrþ,t),
: [0]. Suppose there are at least k nodes along the parental
of a tree of topology TWl, then
parental subnodes for
0i (rþ) moves along
j < k. Clearly, 0o(',þ) :
branch
the parental branch from lþ1,
j
lrþ]. We define the number of internal
nodes along the parent branch of a tree of topology, T , Lo be ,nrl(7). Commencing
at [a(0)] and applying0, a total of N6)
parental branch. Now,
*l
if lrlr]:[0,ir,...,ù^]
times takes us to the leaf node of T's
is some node, ttren
lrll :rrL1-1
is the
depth of that node.
Suppose that the node [T/] is an internal node of a tree of topology
7.
Then the
topology of the tree based around node [T/] is the ordered set,
rr : {[*(rÐ], lrþD@, Ty,þ,tt, T¡4,,rt],
where T¡r,o1is the daughter subtree topology based around the daughter node lrþ,01,
andT¡,¡,,r1is the parental subtree topology based around the parental subnode l',þ,1].
CHAPTER
7, ALGORITHMIC
APPROACHES FOR THE MBT
757
Clearly, then, one can see that the probability of a random tree based around node
[T/] eventually
attaining a topology of T*, as ú --+ oo is given by,
p(Twì
:
pt{ (t'(ú)1,
:
(-Do)-, B (n6¡¡,,q) ø pØr,¡t))
[,tr])(o)
]lPlTþ,0tlPlrw\rl
,
since each event is independent.
The set of internal nodes of an MBT with topolo1y,T, is denoted by ts7. The
set of leaf nodes of that same topology is denoted by, n r and so the set of all nodes
is,
NIz
: ßr U\-r'
Note that we treat the root node, [r(O)] as being an element of
7.3
n
7.
The Depth Algorithm
Let qi be the probability of eventual extinction of an MBT that commences its evo-
lution in phase i from node o(0). The vector, q is then the minimal non-negative
solution of equation (7.7.2), [9], which we write as
s: /(s).
Harris in his seminal
work on branching processes, [9], exploited an iterative scheme to solve the equation
for the probability of eventual extinction for the discrete-time multi-type branching process. In brief then, suppose that qo is any vector in the unit cube of the
appropriate dimension. Then,
:
Jiå /*(qo)
where
.f**r(") : l(lrG)) ir tt"
q,
(7.3.1)
generating function of the offspring probability
distribution for the k + 1-st generation. A similar iterative approach was exploited
by Neuts to solve for G in the level-independent QBD environment. The algorithm
of Neuts can be understood quite simply by a very neat physical interpretation
discussed
as
in Chapter 4. At each step of the algorithm, the set of sample paths that
CHAPTER
7. ALGORITHMIC
APPROACHES FOR THE MBT
are measured are those that commence
in
L(*)
and terminate in
158
L(* -
1) upon
their first visit, such that
1. the maximum attainable level depends on how many iterations have been
performed,
2. there is a restriction on the positions of the left and right transitions, and
3. there is a restriction on how many left transitions are allowed.
Consequently, this space although well characterised, is not easily described.
It is
more difficult to give a physical interpretation for the algorithm of Harris, particu-
larly since q0 can be any vector in the unit cube.
In this section we derive an algorithm that has similarities to both the Harris
algorithm and the Neuts algorithm. \Me call the algorithm the Depth algorithm,
in line with the physical interpretation of equation (7.1.2). The major difference
between the algorithm presented here and the algorithm of Harris is that here we
are dealing with the binary-branch point ctMMTBP, not the discrete-time process.
The interpretation of the Depth algorithm is, as expected, different to that of the
Neuts algorithm. There is however a one-to-one correspondence between the sample
paths of the Neuts algorithm and the binary tree topologies of the Depth algorithm;
we exploit this correspondence in Section 7.3.1 to give a better description of those
sample paths.
In the MBT context, then, we shall implement the following recursion on equarion (7.7.2),
s(0)
(-Do)-'d
s(l)
(-Do)-'d+ (-Do)-' B (s(t-
1)
I
s(l
- r)), t )
r
(7.3.2)
The similarity to the algorithm of Neuts can now be seen directly. However,
a
straight application of the interpretation of the Neuts algorithm is not valid. The
simple left and right transition structure of the level-independent two-dimensional
CHAPTER
7. ALGORITHMIC APPROACHES FOR THE MBT
159
QBD process is not sufficiently rich to physically account for the evolution of the
MBT on the
space of
extinct trees. However, by discarding the level-based physical
interpretation we can give the Depth algorithm a more natural interpretation.
Definition 2 The depth, 6, of a Markouzan
6(r):
bi,nary tree of topology
T
r,s
æß) {l,rrl}
0,1
,1
,0,1,I
,1,1
Figure 7.3.I: An example of an MBT of depth
b.
Figure 7.3.1 gives an example of an MBT whose root node and internal nodes have
been labelled. Its depth, d is therefore,
õ(T):
max{101,10,01,10, 11,10,0, 11,10,1,11,10,0, 1, 11,10,0,1,
1,11}:
b.
We now state and prove a lemma that is important for the correct physical
interpretation of the Depth algorithm. Let T
lT@l
(t) be an evolving MBT, and,
Iet
be the total number of branches at time ú. Then,
Lemma 10 lim¿--
lTØl { N,
almost surely, i,f and, onty i,flim¿*oo
6g(t)) (
oo,
almost surely.
Proof :
Since all states with a non-zero number of branches are transient,
Iim¿-*lf Øl ( ñ, almost surely, if and only if the tree becomes extinct, almost
CHAPTER 7. ALGORITHMIC APPROACHES FOR THE MBT
160
surely. Further, the tree is extinct, in the limit as ú ---+ oo, if and only if on every
branch there are afinite number of nodes and so lim¿--
6g(t)) < oo.
r
As a result of the above lemma, let us denote the space of extinct binary trees
by'lf¡.-.
If
then let ó(*(rþ)) be the phase that the branch (1.(rþ)1, [T/]) was
in immediately following node [*(rþ)). The probability of eventual extinction, Q¿, of
[T/] is a node,
a topology
7r,
whose parent branch,
(["(ú)],
[T/]), commenced in phase ó(*(rþ))
:
¿
has the following physical interpretation,
pllTw,tl.o"l^@þtù:¿l
Q¿
Plõgwì < *lô(,(,Þ)) :,1
(7.3.3)
Thus the probability that an MBT will eventually become extinct is equivalent to
it will have finite depth as ú ---+ oo given that it began in some
ó(*(rþ)) In order to simplify the above equation and those that are of a
the probability that
phase
similar form we shall use an abuse of notation to avoid "storms of subscripts"
[18],
and so we write instead,
Q
:
Pld(rw,l. -lø{"(ø))1
(7.3.4)
So g is just the measure of the space of extinct trees '1f6.-.
Define the sequence q(l) for I
)
1
to be the probability that a tree beginning with
one branch eventually becomes extinct under the taboo that its depth is ð
<I+
7.
Mathematically,
q(t): Pl6Øwì
for all
Iess
<
l+ tló(*(,Þ01,
(7.3.5)
I > 0. In other words q(l) is the probability that a tree will have depth
than I + 1 as ú -> oo, which is equivalent to saying that q(l) is the probability
measure of all extinct trees with ð
< ¿ + 1. Clearly q(0) : (-Do)-'d,
root branch must undergo a catastrophe before a birth.
because the
CHAPTER
7. ALGORITHMIC
APPROACHES FOR THE MBT
Theorem 11 The sequence {q(l)) for I > 0
tonzcallE r,ncreasi,ng and conuerges to the uector
defi,ned bE equati,on
q.
: (-Do)-td
q(t) : (-Do)-'d+ (-Do)-' B (s(t-
161
(7.3.5) 'is mono-
The sequence {q(¿)} also sattsfi,es
q(0)
Proof
:
1)
I
s(l
- r)), t )
r
(7.3.6)
The fact that {q(¿)} is monotonically increasing is obvious. That
verges to q is also obvious since lim¿*oo q(t)
:
it
con-
Iim¿-* PI6(TWì < I + 1ld(*(ri))]
:
ó(.(rþ))l : q.
The next step is to show that {q(l)} actually satisfies equation (7.3.6). A tree
Pl6Ø,þt) <
ool
may eventually become extinct in one of two lvays, either there is a catastrophe at
[ú], ot there is a branch point a,t 11þ] and ([a(Ti)], [',i]) becomes an internal branch.
The two subtrees that are based around [r/,0] and [T/, 1] must then both independently eventually become extinct.
The probability of the first scenario is just given bV
(-ro)-ld.
The second sce-
nario is slightly more complicated. The probability that the root branch, ([r(ú)] ,lrþ]),
will eventually give birth to a daughter branch before it undergoes a catastrophe
is given bV
[r/,0] and
(-Do)-18. The daughter
['rl, 1]
and parental subtrees that are based around
must both eventually become extinct under the taboo that each
subtree has a depth of at most I - 1, so that the entire tree has a depth of at most l.
But the probability of eventual extinction of a tree with at most depth I -
q(l
-
1 is
just
1). Since each of the two subtrees generated by the daughter and the parent
branches are independent, we have that,
q(t): (-Do)-'d+ (-Do)-'B(q(t -
1)
8q(l
-
1))
,
(T.J.T)
and the proof is complete.
The Depth algorithm converges linearly with respect to depth since at each step,
l, of the algorithm, q(l) is the probability measure of all topologies that have a
depth, õ < l. At the next step the space increases by all the tree topologies that
CHAP'I:ER
7. ALGORITHMIC
have a depth, 6
APPROACHES FOR, THE MBT
: I + 1. The number of extra topologies
t62
that are included at
each
step is therefore increased by only a finite number of trees.
7.3.I A New Interpretation for the Sample Paths of the
Neuts Algorithm
The space of sample paths that are included at each step of the Neuts algorithm
cannot be easily characterized, because of the restrictions placed on the number
and positions of the left transitions, see Chapher
4. On the other hand, the set of
topologies that are included at each step of the Depth algorithm is easily described.
For example, at the l-th step, the set of tree topologies consists of all those topologies
of at most depth l.
The similarity between equations (7.3.6) for the MBT and
G(t)
: (- Ar)-' Az + (- A)-1 AoG2 çt - t¡,
(7.3.s)
for the level-independent QBD process suggests that there exists some relationship
between the sample paths of the Neuts algorithm and the tree topologies of the
Depth algorithm. In fact, there is a very intimate relationship: the set of sample
paths measured at each step of the Neuts algorithm can be transformed to the set
of tree topologies that are measured at the equivalent step of the Depth algorithm;
this transformation is one-to-one.
To show this to be true, we must understand the characteristics of the sample
I>
7, record
L(^ -
1) under
paths better. The matrices G(l), given by equation (7.3.8) for all
the probability that a sample path commencing in L(m) will visit
it
L(m+ I + 1), by undergoing at most 2¿ left
transitions. The sample paths that fit the above description also have one more
the taboo that
must remain below
restriction, the position of those left transitions is constrained.
It
equation (7.3.8) is quadratic in G that causes these constraints and
is the fact that
it is these con-
straints that make it difficult to easily describe the sets of sample paths at each step
CHAPTER
7. ALGORITHMIC
APPROACHES FOR THE MBT
163
of the algorithm. For the remainder of this section we shall define and investigate
these conclusions through consideration of the Depth algorithm.
\Ã/e
know for example, that there are an uncountably infinite number of trees that
have the same topology. These trees differ only in the underlying phase process that
has occured along each of their branches. We group these trees together because
their topologies are identical. Similarly, there are an uncountably infinite number
of sample paths that are related by the positions of their left and right transitions.
These sample paths differ because the phase transitions for each sample path are
different.
Let us denote left transitions by L and right transitions by R. As an example,
consider the sample path that is defined by, RLRRLLL. There are an uncountably
infinite number of sample paths that have the above characteristic; they differ only in
their underlying phase process and hence in the length of time until
each
transition
occurs. The probability measure of this set of sample paths is easily seen to be given
by,
(-
Ar)-, Ao(- Ar)-'
Ar(-
Ar)-, Ao(- Ar)-, Ao(- Ar)-'
Ar(-
Ar)-' Az(- Ar)-' Az
Let 3^,^-r denote a set of sample paths from level rn to level m - 7 that differ
only in their underlying phase transitions. Thus, if
S*,*_r:
RLRRLLL, then the
probability measure of this class is given by,
P(S^,^-t)
(- Ar)-' Ao(- Ar)-, A"(- Ar)-, Ao(- Ar)-' Ao
x
(- A1)-r Ar(- At)-' Ar(-
Ar)-t Ar.
In other words the class, S^,*_', ignores all the hidden transitions that may occur.
These transitions are accounted for by the
(-At)-t
term.
Now, let SS,--, be the set of sample path classes that commence in
visit L(m
-
1) that are measured at step
L(m)
I of the algorithm of Neuts. It
show that the number of sample path classes at this step is given by
¡/(l):1+¡/2(l-1),
and
easy to
CHAPTER 7. ALGORITHMIC APPROACHES FOR THE MBT
with ¡/(0)
:
1. This follows directly from equation (7.3.8) and the fact that
G(o)
The spacer Sf,l,--, for all
m
:
(-Ar)-'Ar.
SÍj,l- t
where for each I
)
i e {0,1,. . . , ¡/(¿) - 1}, thus,
: {S'*,,.-r,o,S¿*,^-'t,t,"',Sl.,,^-'r,,nr(z¡-r},
0 we reserve,
single left transition from
foli :
7,.
path classes, one from
L(*) to L(m-1).
. . , ¡f(¿)
s$ll-,
(7.3'9)
Ek,^-r,oto be the sample path class that consists of a
Let Sk,**, denote the sample path class
that consists of a single right transition from
Sl,--r,o
of the level rn. We denote the sample
are independent
path classes from Sl,1J,--, as 31,^-1,,;, where
class
764
- l
L(*)
to
L(m*
1). Every sample path
can be constructed by combining two sample
and the other from s.9,--1, so rer
sf]r,^., e s$ltl-
and Sfi-l-r,k € Sgr)-1, then we write,
sL,^_r,o: s'
where
l)
7,
,**rsli]r,^,¡E''-j^_r*,
j,lce {0,...,¡/(l-1) -1} andi,: l/(l -I)j +(k+t).
(7.3.10)
Thisrep-
resentation is clearly the natural representation when one studies equation (7.3.8).
All of the sample path classes at step l, except the
one sample path class that con-
sists of one left transition, begin with an initial right transition that takes us from
L(*)
to L(m + 1), to which we then append a sample path class that commences
in L(m
f
1) and terminates in L(m). This sample path class has the constraints
and taboos that are imposed at step I
-
1 of the algorithm. To this sample path
class we
finally append a sample path class that commences in L(m) and terminates
in L(m
-
1) that also is under the constraints and taboos imposed upon
it at step
I - 1 of the algorithm; this is exactly as in equation (7.3.10). We choose to order
the sample path classes in such a way that when we combine the 7-th and k-th
sample path classes of the previous iteration we place the new sample path class in
the
positiorr,'i: ¡/(¿ - t)j + (k + t). What
we wish to do, then, is to show that
we can transform the set of these sample paths to the equivalent set of binary tree
topologies whilst also maintaining that same ordering for the binary tree space.
CHAPTER
7.
ALGORTTHMIC APPROACHES FOR THE MBT
165
To begin with, we explain the transformation from the set of sample path classes
to the set of binary trees. There are two types of transitions in the Neuts algorithm,
left and right transitions. There are also two types of transitions in the Depth
algorithm, branch extinctions and internal branch point generation.
It then seems
natural to apply the following transformation,
o left transition --+ branch extinctions, and
o right transition
--+
internal node.
More formally, if the left most unevolved branch is, ([o(/)] ,lrþ])("), then the next
right transition of the Neuts sample path class generates an internal node at ltþ] by
creating a new daughter branch. Immediately after the transition then, we have
lrþl
-
(([*(ø)],
1.,Ä)(o)
,
(rþl,bþ,01)("), ftþl,lrþ,11){")¡,
where (lrþ],lrþ,0])(") utrd (lrþ1,1,þ,1])(") ur" the unevolved daughter and parental
branches. On the other hand if the next transition of the Neuts sample path class
is a left transition, then the branch (l*\Ð],lrþD@ is made extinct. In other words,
([r(ú)],lrþD@
-
([r(ú)], l,þDk)
Let Iú be the transformation that takes us from the space of sample path
to the space of extinct binary trees,
classes
'lfó<oo,
V : S-,--1 -'lfó<To show how we transform from
S-,--1 to 1f6aoo consider the following sample path
class, RRLLRLL. This sample path class consists of two right transitions, followed,
by two left transitions another right transition, and then finally two left transitions.
Let us perform the transformation to this sample path,
{/ ( RRL L RL L )
:
{/ ( R)
so
q/ ( R) ü/ ( L q/ ( L {/ ( R)
)
)
{/ ( L ) {/ ( L ),
CHAPTER
7.
ALGORTTHMIC APPROACHES FOR THE MBT
166
because we apply the transformation to each transition from the sample path indi-
vidually. Applying the transformation to the first right transition we obtain,
(it'{o)1,
[01¡t';r, ([0],
[0,0])("), ([0], 10, 1l)("))ü(R){/(L){/(L){/(R){/(L)\ü(L).
After the completion of the first transformation we are left with a tree that
has
two unevolved branches, ([0], [0,01¡t"l and ([0], [0, t1;t"1. The left most unevolved
branch is ([0], [0,0])("). The next transformation is v(R), and this therefore acts on
branch ([0], [0,0]){"). This right transition generates a branch point at [0,0], and
so
we obtain,
(ttrtolt, [01¡rrl, ([0], [0,0])(''), ([0,0], [0,0,0])("),
([0,0], [0,0, 1])("), ([0], [0, r])t"l;
{/ (L) {/ (L) v (R) {/ (L) {/ (L).
The next transition is a left transition, and we apply this to the left-most unevolved
branch which in this case is ([0,0], [0,0,0])("), and so we get,
({tr{o)1, [01;trl, ([0], [0,0])(t), ([0,0], [0,0,0])("), ([0,0], [0,0, 1])("), ([0], [0, 1])("))
{/ (L)ü (R)ü(L){/ (L).
The next transition is a left transition again, and applying the transformation to
the left-most unevolved branch ([0,0], [0,0, 1])("), we obtain,
({t"{o)1,
[01¡r';1,
([0], [0,0])(¿), ([0, 0], [0,0,0])("), ([0,0], [0,0, 1])("), ([0], [0, 1])(")){/(R){/(L){/(L)
At this point the left
subtree based around node [0,0] is terminated. The left
most unevolved branch is now, ([0], [0,1]){"). To this branch we then apply the
transformation of the next right transition, to get
(tt*rolt,
[0])(n), ([0], [0,0])(¿),
([0,0], [0,0,0])("), ([0,0], [0,0, 1])("), ([0], [0, 1])(¿),
([0,1], [0, 1,0])("), ([0, 1], [0,1,1])(")){/(L)ü(L).
The next transition is also a left transition and we apply the transformation to the
leftmost unevolved branch ([0,1], [0,1,0])(") ¿o obtain,
({tr{o)1, [01;rtl, ([0], [0,0])(t), ([0,0], [0,0,0])("), ([0,0], [0,0, 1])("), ([0], [0, 1])(¿),
([0, 1], [0, 1,0])("), ([0, 1], [0, 1, 1])(r)v(L)
CHAPTER
7, ALGORITHMIC
APPROACHES FOR THE MBT
t67
The final transition is a left transition and we apply the transformation to the last
remaining unevolved branch, ([0,1],[0, 1, 1])("), to obtain,
({t'{o)1,
[01¡ttl, ([0], [0,0])(i), ([0,0], [0,0,0])("), ([0,0], [0,0, 1])("), ([0], [0, 1])(i),
([0, 1], [0, 1,0])("), ([0, 1], [0, 1, 1]),",)
It
is obvious that this transformation is indeed well defined and one-to-one. This
is due to the simple mapping procedure that we apply; each transition has its unique
position. We shall prove shortly that the space Sf?,--, is mapped into the space
of binary tree topologies that have depth 6
< l, which we denote here by 1f6q¿.
This then implies that the space of sample path
classes
at each step of the Neuts
algorithm is transformed into the well described space of binary tree topologies at
the equivalent step of the Depth algorithm.
To begin with
it
is also easy to see that at each step, l, of the Depth algorithm
there are
¡/(¿)
topologies of depth õ
: N"(t-
< I with ,nf(O) :
1)
+1,
1, since equation (7.3.2) is so similar to
1f6q¿
just as we did the sample
{To,r,Tr,r, . . .,T*(r)-r,r},
(7.3.11)
equation (7.3.8). So we can label the topologies in
path classes from S.ÍJJ,--1,
so
1fð.¿
:
where we reserve To,r lo be the trivial single branch topology for all
represent every other topology in
1f6<¿
¿
> 0. We can
by combining two topologies from
1f6a¿-1 at
node [0], so
4,,t:{([t(0)],[01¡tzl,\,t-t,¡0,01,Tr,,-r,¡o,r1],
such that,
(7.3.12)
i,: N(l-1)r+(k+t), and where \,r_r,¡0,o7 undTr,,t_r,¡0,1j are the topologies
of the two trees commencing at [0]. The similarities between, equations (7.3.9) and
(7.3.11), and between equations (7.3.10) and (7.3.72) are striking. We next wish to
prove the following theorem to formalise this relationship.
CHAPTER
7.
Theorem 12
ALGORTTHMTC APPROACHES FOR THE MBT
Il Sl-,,,-t,¿ ts a sample path class of ïfl,*-r,
ú(9j*,_',o)
Proof :
:1,,
€
168
then
1f¿<¿.
To show that,
ú (5,,,*-t ,o) : 4,,
is true, we also use induction. Now, for
I :0 we have that Sf)-- r: {Sk,^-r,o},
therefore,
: {(["(0)], [0])(")] :To,o,
is clearly true for l:0. Suppose it is true for I ü(Så,--,,0)
and so the hypothesis
v@k,L-t,j)
for all
j :0,...,¡/(l -
1)
:
1, that is,
Tj,,-r,
- 1. Clearly, ú(Sj,,--r,o) :
To,,
for all I so we need to
only show that,
V(Si,,--t
foralf
i:1,...,¡/(l) -1.
: v1sl,_*rsl-j r,,,,¡E,_,k_r,)
: v(så,-*r)ú(s'^]r,^,¡)v(sj",l"_r,*)
:
N(l - l)j +
{([o(o)],
[01;t';1,
+ 1) and in the
(k
:1,,,
So,
ü(sj,,__r,o)
where ú :
,o)
\,,-r,¡o,o1,Tr,,-r,to,r]],
(7.3.13)
(7.8.14)
(7.3.1b)
second step we have used the induction
hypothesis. We know that the position of a tree in'1f5<¿ that is formed by combining
the
j-th with the k-th
subtree from 1f¿.¿-r is .n/(l
- I)j + (k + 1) but this is just i,
sot
v (sL,*-
r,o
)
:
{
(
[t
(0)
],
(o),
[0] )
\,t
_.r,¡o,o1,
Tr,, _',to,rt]
and the theorem is proved.
Corollary
:
4,t
(7.3.16)
T
13
ú(s$,--r)
:1fð<¿
(7.3.17)
CHAPTER
7.
Proof :
Follows immediately from Theorem 72.
ALGORITHMTC APPROACHES FOR THE MBT
\Me have now shown
169
that there exists a one-to-one correspondence between the
sample path classes of the Neuts algorithm and the binary tree topologies of the
Depth algorithm. Further, at each step, k, we can map the
that step to the ,n/(k) binary trees that are of depth
l/(k)
sample paths of
less than or equal
to k
and
therefore give an easily identifiable description to the sample paths of the Neuts
algorithm in terms of binary tree topology.
In Section 7.5,, an algorithm that is analogous to the QBD algorithm [/ is developed. \Me call this algorithm the Order algorithm. There is no counterpart to
this algorithm in the branching process domain. We shall
see
that the Order algo-
rithm converges at a faster rate than the Depth algorithm, in the same way that the
algorithm
7.4
[/
converges at a faster rate than the algorithm of Neuts.
The Order of an MBT: Definition
Definition 3 The order of a srngle branch zs 0. The order of an'internal
0r(rþ), on the parental branch of a topologU
T¡,l,7,
is
denoted,
node,
by Øn(ïr(rþ)) and,
zs
gr,uen by,
a" (or (lþ))
:
a
(T¡rr
çE¡,01),
fork:0, 1,2,...,Nø,þ) . Fi,nally, the order of Tr i,s gi,uenby,
1+k:O,I,2,...,N(T"þ)
max
a.(or (rþ))
Fþom the definition
it is clear that we need to calculate the order of an MBT
recursively. We start with all the nodes of the parental branch and then work our
way down the subtrees until we reach the leaf branches. The order of the tree is
then 1 more than the highest order node along the parental branch. Consider the
tree that is depicted in Figure 7.4.I. Let the topology of the entire tree be denoted
by,
T.
The parental branch of
T
has only two internal nodes along the parental
CHAPTER
7, ALGORITHMIC
APPROACHES FOR THE MBT
170
branch, and they are, [0] and [0,1]. Let us calculate O,,(0,1) first,
t)
:
O,(0, 1,0)
:
O,(0,
0(Z¡o,r,o1),
and
O(7¡6,r,q)
:
1
I
1
+
O(T¡o,i,o,ol)
:
1
f 0:
1,
since 7¡o,r,o,o1 is a single branch and hence has order zero. So,
O,(0, 1) :
O(Zto,r,o1)
:
1.
T
T
T
t0,11
r\0r
1l
t0,0,01--------*
1,0
T
--)Trot,t
[0,1,0,1]
T
1,01
[0,0,0,1,1]
t0,1,0,01
Figure 7.4.7: An example of an order calculation.
Let us now calculate O"(0). This is a little more difficult due to the more complex
natrrre of the subtree, T¡o,s1. Let us go through the calculation. Now,
O"(0)
:
O(Z¡0,01),
and then,
O(7¡6,q)
:1*O"(0,0),
since 7¡¡,01 has only one internal parental node. However
O,,(0, O)
:
O(7¡s,o,o¡),
CHAPTER
7. ALGORITHMIC
APPROACHES FOR THE MBT
177
and since T¡o,o,o1is a subtree with two internal parental nodes, we know that,
O(T¡s,o,q)
It
: 1*
max{O"(0,
O,(0,
0, 0),
0, 0,
1)}.
is clear that
O,,(0,0,0, 1)
:
O(7¡o,o,o,r,o1)
:
0,
since this subtree consists of only a single branch. In addition, we have that,
:
O,(0,0,0)
0(7¡6,6,¡,01)
:0,
since this subtree also consists of only one branch. Therefore, we have that,
o(T¡e,o,q)
:
1
* max{o"(0,0,0),o,(0,0,0, 1)} : 1 * 0 :
1.
Consequently, then,
O",(0,0)
:
O(7to,o,o1)
:1,
and thus,
O(7¡s,q)
:
1 -l-
O",(0,0)
: t | 1:
2,
resulting in,
O"(0) :0(7¡6,¡1)
:2
Finally then, we have that,
O(7) :
1
+max{O"(O),O,(0, 1)}
: 1f
max{2, 1}
:
7
l2 :
3
So the order of this particular tree is therefore three.
Note that trees from the same topologically isomorphic class can have different
orders. By rotating uneven branch points, the nodes of the parental branch change
and hence the daughter subtrees also change. Since order is calculated with respect
to the parental branch and its internal nodes, the calculation of the order may
therefore be different for topologically isomorphic trees.
The number of topologies that are of order I for all I
>
1 is
infinite. One can see
this, by understanding that if the orders of all the nodes of a tree of order
I
are at
CHAPTER
7. ALGORITHMIC
APPROACHES FOR THE MBT
772
/¡
Figure 7.4.2: Two different trees of order one.
most I - 1 then we can construct a tree from an infinite number of building blocks of
order at most I - 1. Figure 7.4.2 depicts two trees of order one. The second topology
has twenty internal parental nodes each of order 0, whereas the first topology has
only two.
Lemma 14 lim¿*-
lf @l (
oo, almost surely, i,f and, onty i,f A(T(¿)) < æ, almost
surely.
Proof :
Since all the states with a non-zero number of branches is transient,
| a -, almost surely, if and only if the tree has become
extinct, almost surely. F\rrther, the tree is extinct if and only if on every branch
we have that lim¿-
""lf
çt¡
there are a finite number of nodes and so, 0(7(¿))
<
oo as ú --+ oo, almost surely.
CHAPTER
7.5
7. ALGORITHMIC
APPROACHES FOR THE MBT
L73
The Order Algorithm
We saw in the previous section that there are an infinite number of topologies
that are of order
I > I, whereas for depth I > I
of topologies. As a result,
topologies of order
if
there are only a finite number
we could devise an algorithm that includes all the
k < I at step l,
then we would have a much more efficient
algorithm than the Depth algorithm. The Order algorithm that we discuss in this
section is just that algorithm.
The Order algorithm is analogous to algorithm
[/ from Chapter 4. We begin by
re-writing equation (7.L.2) as,
s: (-Do)-rd+ (-Do)-tB(s
ø r{1))s.
(7.5.1)
If we substitute this equation into itself we obtain,
s
where
: (-Do)-'d+ (-Do)-tB(s 81(1))(-r0)-Ld+ ?nù-ts(s
: (-Do)-'d+ x(-Do)-rd ¡ x2 s,
ø l{1))2s
X : (- Do)-tB(s ø f{t)¡. If we now repeat this substitution I times we
obtain,
¿-1.
": Ik:o xr?nù-'a+ n,çs¡,
where A¿(s)
(7.5.2)
: Xts is the remainder term. Now if we take the limit as I ---+ oo we
obtain,
":Ëk:o xrGnù-1d+R(s),
where
A(s) :
Iim¿-oo Æ¿(s). The above expression
(7.b.3)
is well defined because s
is
a probability. However, the remainder term may be non-zero. Consider now the
minimal non-negative solution to equation (7.7.2), q. Now, q must also be the
minimal non-negative solution to equation (7.5.3), so we write,
oo
q: t ur?nù-'a+ a(q¡,
Ic:O
(7.5.4)
CHAPTER
where U
7. ALGORITHMIC
APPROACHES FOR THE MBT
774
: (-Do)-tB(ø ø fttl¡. It is a well known fact from branching process
theory that g is the probability measure of all sample paths that eventually have zero
living particles. Due to transience and regularity this probability is the same as the
probability of all the topologies that consist of a finite number of branches
as ú
---+
oo.
The first term of equation (7.5.4) is the probability measure of these topologies, and
consequently the second term is the probability measure of all those topologies that
eventually become extinct after having an infinite number of branches. However, we
know from branching process theory that this occurs only on a set of measure zero,
and hence the second term, A(s)
:
O,
for the physically significant solution,
s:
q.
We therefore consider the expression
"
: Ë (i-ro)-'r(s ø rtrr¡)r eoù-'o
k:o
(7.b.b)
The form of equation (7.5.5) has a very interesting physical interpretation, an inter-
pretation that allows us to develop the Order algorithm.
Let us begin by analyzing,
U
: (-Do)-t g(q 8 1(t))
at a node, say [a(/)], and suppose that eventually an observable
event occurs at node [ú], ro that node [r/] becomes an internal node. The daughter
Suppose we are
branch at node [T/] evolves into a subtree that eventually becomes extinct with topolo1y, T¡¡,o1. For the purposes of the above expression, whilst the daughter strbtree
is evolving towards extinction, the parental branch, (llþl,lrþ,1]){")
¡
unevolved until
the daughter subtree has become extinct with that topology. We call the entity that
is based around [T/], that has representation,
{fl"(ú)l ,lrþD@ ,Ttþ,ot, (rþ1, [,/, 11¡t"11
a U-unit. Note that A"(rþ)
(
oo since lfW,rtl
< oo. The probability of a(I-unit,U,
is given by the product of the probability of the initial branch point, (-Do)-18, the
CHAPTER 7. ALGORITHMIC APPROACHES FOR THE MBT
775
eventual extinction of the daughter subtree, q and the suspension of the parental
branch ¡(t), and since each event is independent we have,
U: (-Do)-'B(qel1(t)).
Figure 7.5.1 depicts an example of a [/-unit; the arrow on the parent branch imo(e
qJ
Figure 7.5.7: An example of a U-unit.
mediately following [t/], indicates that the parent branch is suspended, until the
daughter subtree has become extinct.
Any extinct tree can be constructed by combining a finite number of [/-units
together; connecting the unevolved parental subnode lrþ,t]t"¡ of the previous [/-
unit to the parent root node of the next [/-unit. Finally, the tree is terminated
by a catastrophic transition on the unevolved parental sub-node of the parental
branch of the entire topology. For example,Uk(-Do)-td is the probability that a
tree becomes extinct with a topology that is constructed from k [/-units combined
together, followed by a catastrophe; hence the parental branch has k internal nodes.
Therefore,
q: Ë ukeDù-td,
k:L
gives us the probability of the set of extinct trees consisting of a finite number of
U-units, or a finite number of internal nodes along the parental branch. Figure 7.5.2
CHAPTER
7. ALGORITHMIC
APPROACHES FOR THE MBT
176
U
U
Figure 7.5.2: An example of an extinct tree built from four U-units
depicts a tree that is constructed from 4 U-tnils; a tree of this type is an element
of the space with probability of
u4(-Do)-'d.
Let B[7k(tþ)] be the event that a parental branch commencing at node þ has
undergone k observable transitions since the node lþ1. Let Al0r (rþ)l be the event that
the parental branch has undergone k - 1 observable transitions since [T/] followed by
acatastrophictransitionat l0r(rþ)l Letóo(tþ) givethephaseof theparentalbranch
immediately after node [r/], in other words the initial phase of the branch (lrþl,l',þ,11)
Thus, óo@r(rþ)) is the phase of the parental branch immediately following node
l0r(rþ)| in other words, the initial phase of the branch (10*(rþ)l,l0r*'(rþ)]) for aII k
such that the parental branch has at least k parental subnodes.
Definition
U¿j
4
:
The
matri,nU:[U¡¡]
Plß10(1þ)1,
fori.,i:1,2.,...,n
A"(0(rþ)) < oo & ór(0(rþÐ
:
i,s defi,nedto be
jlóo(rþ)
: il.
(7.5.6)
We stress that U is independent of the position of the initial node. However, the
matrix U is dependent upon the initial phase of the parent branch, óoþþ), and on
the phase of the parental branch immediately following 0(rþ), ór(0(rþ)). Once again,
CHAPTER 7, ALGORITHMIC APPROACHES FOR THE MBT
as an abuse of notation we shall also write
U
:
U
t77
as,
P[ß10('r)], O" (0(rþ)) < oo & ór(0(rþDlóoþÐ),
(7.5.7)
in order to avoid the overuse of subscripts.
Definition 5
The uector
q
co,n be defined as,
Q: Plo(rrúr) < -lø("(ø))]
(7.5.8)
The Order algorithm to determine the minimal non-negative solution to equation
(7.5.5), and hence (7.7.2), is
q(0)
:
U(I)
: (-Do)-'B(q(t-
q(t)
:
(-Do)-td,
(7.5.e)
ør{t)¡ ,t>
1)
7
(7.5.10)
1
(7.5.11)
oo
ttu(l)l¿(-Do)-'d,
t:0
Theorem 15 The sequences {U(l), ¿ > 1} and{q(l), ¿ > 0} definedby
u(t)
:
pIß10(1þ)1, a"@?þ))
<t
k ó"(0('þDló,(rþ)1,
(7.5.12)
and
q(t): p[avwì
sati,sfy
<t+11ó(o(,þ))],
(7.5.9)-(7.5.11). The two sequences are monotoni,cally'increas'ing and respec-
ti,uelg conuerge to the
Proof :
matrir U and the uector q.
We first show that the sequences
I/(l)
and q(l) defined by (7.5.12)
[/
and g respectively. Tlivially,
and (7.5.13) monotonically increase and converge to
{y(l)}
(7.5.13)
and
{q(l)}
are monotonically increasing. F\rrther,
,tlt ¿tt,l
:
:
Iim Plß10(rÐ1, a"@(,i )) < t k, ôe@(1þDlór(rþ)l
PlBl0(1þ)1, a"(o(rþ))
rT
u1
< oo & ór(0(rþÐlOrþt
l
CHAPTER
7, ALGORITHMIC
APPROACHES FOR THE MBT
178
â,nd
,t1iø(t)
:
rim P[o(Z) < I + 1ld(CI(o))]
:
Plag) < -lø{"(o))1
q
The matrixU(l) gives the probability that beginning at some node, [ú] itr phase
ór(rþ),
a
branch point eventually occurs at the parental subnode, l0(rþ)1, the daughter
subtree that is based around l0(rþ),0] has order at most
l-1,
so
that,0,,(á(rþ)) <1,
and the parent branch at l0(rþ)) is in phase ór(e(rþ)). Therefore, we have,
u(t)
: plß10(1þ)1, a"@þþ)) < r k ó,@þþDlü,rql).
However, the condition A"@(þ))
< I is equivalent to saying that
O(7tú,01)
3.5.14)
( l, so
we can write,
u(t):
plß10(1þ)1, o(7¡a,01)
<t k ó"(0(''þDlo,rlù1.
(7.5.15)
Since each event occurs independently, we can re-write the above equation as,
u(t)
: t
t
plftþ1,[á(d,)])(,) uó,(0(,þ))
ut(.ftt,01))ld"(,i)l
ôr(0(,þ)) ó(a(,þ,o))
xrlaQ*,ol) < rlø('(t'l, ol))l
The event ßleþÐ] tells us that a branch point eventually occurs at node l0(1Ðl@
with the daughter branch in phase ó(t(lrþ,0]) and the parental branch in
phase
lór(0(rþDl immediately after the branch point, the probability of this is given by,
plftþ1,10(1ÐD{"Ð e.
ó,(e(Ð) u o(*(,i,01))10"røll
which is just
(-Do)-'8. rhe
term P[O(7¡a,01) < tló(a([ø, o]))] is just the probability of the tree based at [T/,0]
becoming extinct with order at most I
-
1, which is given bV q(l
-
1). Therefore we
obtain
u(t): ?nù-'n(q(t -
1) ø
r{t);,
(7.b.16)
where the Kronecker product with 1(1) represents that the present branch is frozen
with probability 1.
CHAPTER 7, ALGORITHMIC APPROACHES FOR THE MBT
179
Let Tfi, be a topology that is based around node [T/] whose parental branch
has exactly k internal nodes before terminating. The first internal parental branch
subnode is of course lrþ1. Suppose
that this topology has, O(7'f;¡) < Z +
t.
The
probability that a random tree will eventually have this property is just,
Plagtì
It
< ¿+
rló(a(,þ))l
(7.5.17)
is not difficult to see that the above expression is equivalent to,
plAllk(,þ)l s,agfþ) <t+
since there are k
- l
tlø("(ú))1,
(7.b.1s)
internal nodes from [T/], the first internal node. At the k-th
parental subnode from the parent node [T/], a catastrophic event occurs. Now, the
order of a tree is one more than the order of the node on the parental branch with
the highest order, and
so,
:
PlAllk (,þ)l k
a"(0'(,,þ)) < rló(*(lù)).
o:il_1ä_,
kagtþ) <t+ rlø("(ú))J
plAl7k(,,þ)l
(7.5.1e)
The expression in equation (7.5.19) is identical to saying that the orders of each of
the nodes must all be less than l, so we have,
plAl7k (,þ)l
k,:il?f
plAllk (,,/)], o"(ú)
.
(,þ)) < tl4(ai,þ))l
,Ø.(e'
t,a"(0(1þ))
.
l,.
. . , o,
:
(0r-'(,þ)) < tló("(,Ð)1.
Now each node and hence daughter subtree evolves independently, so,
p
lAllk
t
óp(rþ)
(ø) l,
t
o" (ø) )
.
t,
a"
(1þ))
.
t, . . .,
o, (0r-' (lþ)) < tlô (a(1þ))l
plß?þ|o"('/)) <t kó,(,,þ))lø("føll)l
ór(0k-1(rþ))
X
r
x
rlnlek-r (ú)], o" (0r-'(rþ))
lnle
(0
(t¡;)1,
a"(0 (,þ)) <
t, k
x eltlek (þ)lló"(er-' (,þ))).
ó"(0 (,þ)) ló,(,þ))l
{ t, k óo@*-'(,þ))lór(0*-'(rÐ)l
CHAPTER
7. ALGORITHMIC
APPROACHES FOR ]:HE MBT
The first k terms are each just terms of the form of
equal
to (-D6)-1d,
because after the
t/(l)
180
and the last term is just
k-th internal node of the parental branch,
a
catastrophe occurs. In other words,
Plagtì
< t + rló(a(,þ))l
However, to obtain q(l) we must sum over all
:
uk
(D?
k,lhal
Do)-' d.
(7.5.20)
is, over all the possible number
of internal branch points along the parental branch, so,
oo
\
q(t)
r¡açr6r) < I + 1 ld(û(ú))l
Ic:O
oo
(7.5.21)
Ður(t)(-Do)-'d.
k:0
7.6
Comparing the Depth and Order Algorithms
In the levei-independent QBD
to level and
it
case,
the algorithm U converges linearly with respect
converges at a faster rate than the Neuts algorithm. The reason for
this is that all the sample paths included in each iteration I of the Neuts algorithm
are also included at the
l-th iteration of the algorithm [/. However, the algorithm
[/ does not place a constraint on the number or pattern of left transitions, unlike
the Neuts algorithm and hence includes many more sample paths.
I
z_
2_
Figure 7.6.L: The space of trees included at the second iteration of the Depth
algorithm, with their order also indicated
CHAPTER
It
7.
ALGORITHMTC APPROACHES FOR THE MBT
181
is interesting to consider whether there is a similar relationship between the
Depth and Order algorithms. Figure 7.6.1 illustrates all the topologies of the trees
represented at the second iteration of the Depth algorithm. The order of each tree
has also been indicated. The maximum order of the tree topologies at the second
iteration is two. This illustrates a more general property.
Theorem 16 The
hi,ghest order tree
tn the l-th'iterati,on of the Depth algori,thm
r,s
t.
Proof :
The proof is by induction.
At the zeroth iteration of the Depth
algorithm, oniy the single branch is included, and by definition the order of a single
branch is zero. Hence the statement is true.
Suppose
it
is true for the l-th iteration, that is, the maximum order tree is
l. At
the I * l-st iteration of the Depth algorithm, the space of trees is constructed by
combining two trees from the l-th step at a branch point; the daughter and parental
subtrees. By the induction hypothesis, the daughter and parental subtrees are of
order at most
L The order of the combined tree is 1 plus the order of the daughter
subtree with the highest order, and this subtree is clearly the above daughter subtree.
Therefore the order of the tree is at most 7 +
l.
Thus the maximum order at the
r
l+l-thstepis l+7.
The l-th iteration of the Order algorithm includes all trees of order at most
without any restriction on the depth. Hence
it
I
includes all of the trees that are
included at the l-th iteration of the Depth algorithm. Thus the Order algorithm
converges more rapidly than the Depth algorithm. Figure 7.6.2 depicts a tree that
appears at the first iteration of the Order algorithm, but not until the 20-lh iteration
of the Depth algorithm.
CHAPTER
7.
ALGORTTHMIC APPROACHES FOR THE MBT
t82
Figure 7.6.2: A tree of order 1 that only appears at the 20-fh iteration of the Depth
algorithm.
7.6.I
Numerical Comparison of the Depth and Order Algo-
rithms
Consider an MBT with
Do:
100
0-1
00-1
0
(7.6.1)
CHAPTER
7. ALGORITHMIC
APPROACHES FOR THE MBT
183
and
Bwhere 0
l-e 0 0 0 0 0 0 0 0
0
0 000.5000
0 000 0 0000.5
( e < 0.5. Figure 7.6.3 compares
(7.6.2)
the average CPU times for the Depth and
Order algorithms using the above simple MBT. The algorithms were both ran one
hundred times. As can be seen from the flgure, the Order algorithm does indeed
outperform the Depth algorithm, Notice how as e approaches 0.5 both algorithms
CPU Time versus epsilon for the Depth and Order Algorithms
101
- - -
Depth
Order
o0
o
E
tr
l
fL
()
o,
o
-J
1
0
-c
10-
0
0.05 0.1
0.15
0.2
0.25
0.3
0.35 0.4 0.45
0.5
Epsilon
Figure 7.6.3: Comparison of the Depth and Order algorithms as e varies from 0 to
0.5.
take considerably longer to determine the probability of eventual extinction, since
the dominant eigenvalue of Do-lBC approaches 0 and the process becomes critical.
CHAPTER
7.7
7. ALGORITHMIC
APPROACHES FOR THE MBT
184
Logarithmic Reduction Algorithms
The Neuts algorithm and the algorithm
[/ were developed to find G in the level-
independent QBD domain because each of the non-absorbing levels are identical.
These algorithms can be interpreted physically
process and the sample paths
lry analyzing the levels of the QBD
in and between these levels. The MBT on the other
hand, is a level-dependent QBD and
it
is rather striking that essentially level-
independent algorithms can be developed. That these exist is a consequence of
the special higher level transition structure of the MBT. This transition structure
is based on the fact that there is no interaction between any of the living branches
of an MBT; they evolve independently. This means that each branch of an MBT
can be isolated and allowed to evolve whilst all the others are suspended.
independence of branch evolution
It
is the
that allows algorithms anaiogous to the Neuts
algorithm and the algorithm U to be applied successfully, despite the fact that they
are essentially level-independent algorithms.
Since the Depth and Order algorithms converge linearly with respect to depth
and order we wish to study whether there are algorithms that converge at a faster
rate that can be implemented in the MBT regime. Algorithms for analyzing QBDs
that converge quadratically with respect to level have already been suggested. For
example, the logarithmic reduction algorithms [3, 17, 28] are in this class.
It
would
be desirable to develop a quadratically convergent algorithm with respect to some
quantity, such as level, or a more generalized order concept in the MBT regime. We
shall investigate the level-independent and level-dependent logarithmic reduction
algorithms each in turn.
The level-independent logarithmic reduction algorithm (LILRA) can be found in
[17, 18] and we discussed
it in Chapter 4. Define A(l) to be the complexity
level of
e {0,I,2,. . .}. W" call C the complexity level of the tree because we do
not specify whether it is the level in a QBD sense, or another concept such as order
a tree for I
or depth as defined earlier. Furthermore, define 1(l) to be the first passage time
CHAPTER
7. ALGORITHMIC
I
into complexity level
for I
APPROACHES FOR THE MBT
:0,!,2,.
.
., that is, 7(l)
:
185
inf{ú > OlX(ú) € A(¿)}.
The matri* ¡¡[n) has the form,
¡¡lk)
:
¡¡tn) trlt<)
¡
(7.7.1)
¡l*) ¡1ln) ,
where
¡[tkt
:
pdQr*') < ?(0), k, x(1Qk+'))lx(0) € a(2k)l ,
(7.7.2)
and
trtk)
:
plz(o) < 1(2r*'), & x(7(o))lx(o) € a(2k)1.
Suppose we are currently
(7.7.3)
in complexity level, C(Zr+t;. Flom this complei<ity level,
applying the operat or Hlk) takes us to complexity level C(2k+' +Zr) and then applying the tr[k] operator takes us down to complexity level C(Zr+t¡. On the other hand,
if we apply operator,
,Llkl
first we flnd that we go down to C(2k), and then applying
operator fIlk] takes us to level C(2k+1). By the mere complexity with which trees
evolve, the manner in which the tree traverses from C(2k+t
+Zr) to C(zr+t; must be
different from the way in which the tree traverses from C(2k) to C(2k+t). kr other
words the two uses of the term ,L[k] require different expressions and similarly the
two uses of Hlkl aho require different expressions. Consequently equation (7.7.1) is
not valid for the MBT, and thus the level-independent algorithm is not applicable.
The levei-dependent logarithmic reduction algorithm (LDLRA) [3, 28] can, of
course, be applied
to the QBD representation of the MBT. It is appealing since it
is known to converge quadratically with respect to level. So whether the LDLRA
should be applied depends on its overall efficiency when compared to the linearly
convergent Order algorithm. Recall from Chapters 4 and 5 that the number of
phases in the level-dependent QBD representation of the MBT at level I is denoted
Tty
M¿: r¿1, where there are n distinct particle types.
At the k-th iteration, the Order algorithm has complexily O(n3) whereas the
LDLRA has complexity O(M2k-'M2rMs). Now in the MBT
case
this means that
the process has complexity,
t) (n2r-'
n2n)
,
(T
.7.4)
CHAPTER 7. ALGORITHMIC APPROACHES FOR THE MBT
since there ar.n"*-'phases in
186
L(2k-r), n2* phases\n L(2k) and one phase in f(0)
This equation can be simplified to give,
O(n3(zk-'l¡
Suppose then,
:
that n
(7.7.5)
2 and suppose that one thousand iterations of the Order
algorithm are required to obtain an acceptabie degree of accuracy, whereas only
10
of the LDLRA are required to obtain that same accuracy. Now the Order algorithm
requires approximat ely O (23) calculations each iteration and there are one thousand
iterations, so we have approximately eight thousand calculations. The LDLRA
algorithm requires approximately, O((zsfz')))
-
21'536
calculations just for the tenth
iteration. Clearly, what the LDLRA gains in convergence properties
it
loses
in the
number of calculations required at each iteration because of the massively increased
size of the matrices involved in the calculation. Thus, despite the Order algorithm
converging linearly,
it
is still more efficient than the LDLRA in most
cases.
We could devise an algorithm based on the alternative state space representation
given
in Section 5.1.1, that is the representation where we count the number of
branches in each of the phases. This representation has a smaller state space than the
conventional representation we have employed throughout. Might not an algorithm
based on this representation perform better than the Order algorithm? In this case,
then, we have that,
O(M2r-'M2xMs)
Now suppose that n
('r-';r:-
:
:
2k-r+n-l
2k-1
X
2k+n-7
2k
2 and say that we need ten iterations, then we require
t)
('-
+;- t) :
Qs
+t)Q,o + t) :525825'
(7 7 6)
calculations just for the tenth iteration. Therefore the LDLRA based on this repre-
sentation still seems to be more inefficient than the Order algorithm in most
cases)
although it would perform better than the LDLRA based on the original state space
representation.
CHAPTER 7. ALGORITHMIC APPROACHES FOR THE MBT
t87
The algorithms that have been successfully developed for the MBT have been
interpreted using specific measures of complexity, for example, the depth of the tree
and the order of the tree. The LDLRA, which has a physical interpretation that is
based on the level of the QBD, can be directly applied to the
MBT. However such
an approach yields matrices that become extremely large even at early stages of
the algorithm and hence is not feasible. Another avenue for developing algorithms
that converge faster than the Order algorithm is in generating different measures of
complexity and finding algorithms with linear convergence with respect to the new
measure. We have attempted such an approach, but we believe that these algorithms
suffer from a flaw that is similar to the LDLRA flaw, namely, that they are plagued
by matrices that at each iteration become progressively larger. The reason for this
is that the subtree units, with which the trees are built at intermediate steps, will
require multiple branches to be concurrently alive. Whilst we cannot rule out the
possibility of the existence of such a measure of complexity, our experience points to
no such measure, and thus no other algorithms that converge faster than the Order
algorithm.
Chapter
I
The General Markovian Tree
8.1 Introduction
We have shown in the previous two chapters that the binary-branch point cIMMTBP
when represented in the MBT format lends itself easily to algorithmic analysis. The
next step in the process is to re-write the general ctMMTBP in terms of a struc-
ture similar to that of the MBT. This will confer to the cIMMTBP an excellent
foundation from which to begin analysing the process algorithmically. The general
Markovian tree, (MT), which forms the basis of this chapter, provides that alterna-
tive representation. This representation of the general ctMMTBP as a Markovian
tree enables a field that is almost devoid of algorithmic approaches [S] to become
subject to powerful matrix-analytic techniques. The alternative interpretation of
hidden and observable transitions of the Markovian tree also provides a natural step
towards developing a cIMMTBP structure that has correlations between the parent
lifetime and the offspring distribution.
In Section 8.2 we define the MT representation. In Section 8.3 the equivalent
cIMMTBP representation is stated. In Section 8.4 a general Markovian tree labelling system is introduced. In Section 8.5 the Depth algorithm is discussed. The
Depth algorithm is equivalent to the Harris algorithm for the discrete-time multi-
188
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
189
type branching process [9], the difference being however, that we have given this
algorithm a novel and interesting physical interpretation. We define the order of a
Markovian tree in Section 8.6 and then finally, in Section 8.7 we discuss the Order
algorithm for the MT.
8.2 The Markovian Tree: Definition
The Markovian tree is a level dependent process with states,
x (t) : (¡r(¿), ór(t),. . ., dr,,1a(r))
UËo{{f} t {1,... ,r}^}.The random variable, ¡tr(¿),
denotes the number of living branches at time I and the random variables, ór(t),
for ali k e {1,. . . , ¡rr(ú)} denote the phase of the k-th branch at time t. LeI N^ be
the rn-fold Cartesian product of ,A/ : {7,2,,...,n}, for m > 1. The level of an MT
defined on the state spac"
is given by the number of branches that are alive; suppose that there ate rn living
branches, we denote the level by
L(m) for m € {0} UZ+
L(*) : {(^, ór,.. ., óòl(ór,..
At L(m) there are n^
,
., ó,,) €
N^}.
possible states. The level, ,C(0), is populated by the state
with zero branches, that
is,
L(o)
: {(o)}
The transition rate matrix for the process is defined to be
Q:
0
0
0
Aqì
A5')
A\Ð
o
o
0
0
At') A!:)
A?] Ar) A\') Ay)
0
Ag),
As',)
Af)
Since the Q-matrix for the process is conservative we have that
D
rn:-l
tl!,)
: o,
(s.2.1)
CHAPTER
for all k
8.
> I.
THE GENERAL MARKOVIAN TREE
190
Note that we have changed the notation of the
Chapter 5. The reason for this is that lhe m
A
matrices from
n A*) now refers to the number of
{-7,0,1,. ..}, whereas
utilized. The reason for this
new branches that are formed at a branch point, for all rn e
in Chapter 5 the standard QBD nomenclature
was
change in nomenclature stems from the fact that we can now have more than one
new branch emanating from a branch point and thus the process is no longer a QBD
process.
The nk x nk matrices A[k) are given by,
Ay)
and A[o)
:
:
Af-t¡ o Do, for k )
1,
0. Where the matrix D¡ has the property that
(8.2.2)
(Ds)¿¿
(
0 and (Do)¿,
>
0
for 1 ( i.+
j < n. The interpretation of A5*) ir identical to that of equation (5.1.2)
from Chapter 5. The matri""t Afì are of dimension ntr x nk-1' and are given by
k-7
AYI:D,t'u cld8 l(k-t-t), k>
7
(8 2.3)
j:o
:1,
k) 1¡(n) arethe nkxnk identitymatrices. Thenx 1
vector d has components d¿ ) 0 for 1 < 1 < n with at least one component being
where¡(0)
and for
strictly greater than zero. The matri"". ,4f] have interpretations identical to those
of equation (5.1.1) of Chapter 5. The
A*) matrices are of dimensionnk xnkt*
for
m) I and can be expressed as,
k-r
A*)
-_
Ðtr¡
j:o
Ø
B*6 ¡(t-t-i) , k ) 7.
(8.2.4)
The element (B^)¿,¡o^...¿n_tim gives the rate at which a branch point occurs such
that immediately after birth the parental branch will be in phase i,^ and the rn new
,i*_1, given that the parent branch
was in phase i immediately before the birth. Therefore, A#) gives the total rate
daughter branches will be in phases, 'io,'it,.
..
at which a single branch from the k possible branches will give rise to m daughter
branches in the one transition.
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
191
This alternative representation of the cIMMTBP in terms of an MT enables
a
different interpretation. This interpretation results from the distinction between
observable (non-singular) transitions and hidden (singular) transitions. We are not
concerned
with particles but instead consider branches and the phase
processes
acting on these branches. The phase process generates the correlations that are
possible between branch lifetimes, which now may be non-exponential, and the
phases of the daughter b,ranches at their birth.
At each branch point of the MBT,
we designated the daughter branch to be the
Ieft branch and the parental branch to be the right branch. We saw the significance
of this in Chapter 7 because this particular orientation allowed for a natural phys-
ical interpretation of the resulting algorithms. The purpose of this chapter is to
provide numerical algorithms for the general Markovian tree, and as such a similar
designation is required. In this case, the right most branch at each branch point is
chosen
to be the parental branch regardless of how many daughter branches
have
also been spawned.
Suppose
that at time ú the process is in a state with M branches and let branch
k < M be in phase
r.
LeI the current state of the process therefore be,
(M,
b, r) c)
k-7 k k+7
0')
1
d)
M
where each branch is labelled by the number below that branch. The following
transitions are then possible:
o A hidden transition to
causes
phase
j I r, occurs with rate (Do),¡.This transition
the state of the MT to become
(M,
0,)
1
b, i, c)
k-L k k+L
d)
M
o An observable transition that generates rn branches for m
) I.
Such an
observable transition occurs with rate (B^)r,¿o¿r...i*_ti^. As stated above, we
CHAPTER
8. THE GENERAL MARKOVIAN
TREE
792
orient the tree such that the right most branch is the parental branch. The
new state of the MT is
(M
+m,
b,
k-7
a)
1
,io,
Lm-|¡
Lm¡
k
k+m-l
lc+m lc+m+7
where the daughter branches'is,'i1,...,,i^-t are designated
the parental branch is now the k
previously labelled
k+L,.. ., M
*
d)
c1
k,...,k t
cease
m
-
L,
m-th branch and the branches that were
have been re-labelled to
k+m|l,
o Finally with rate d" a catastrophe occurs on branch k. This
to
M+m
. . .,
M +m.
causes branch k
to exist and the new state is
(M
- l, 0,) .. .)
b,
c,) .. .)
1 ... k-7 k
The branches that were previously labelled
d)
M-r
k+1,..., M have been re-labelled
tok,...,M-7.
8.3 The Markovian Tlee: ctMMTBP Representation
8.3.1 Definition
Let /(s)
:
(,fttl(s), ¡(z)("),.
..,
f(") (s)) be the generating function of the offspring
probability distribution for the Markovian tree. Let ø be an n x
I
vector, and let
æ(*) denote the m-fold Kronecker product of the vector æ. In other words, æ(-) is
defined by,
æ(*)
with
æ(1)
:
- *(*-r) Ø æ,
(S.3.1)
Ø. \Me define the vector, â, to b.,
âo:
.,
-*
-(Do)oo'
(s.3.2)
CHAPTER
for all 'i :
8.
THE GENERAL MAR.KOVIAN TR.EE
7, . . . ,tu,
193
the matrix, D6, to be
(?:)nl
(Ðo);t:
.
?'r
(8.3.3)
-(Do)ui
forl ( i,+ j <nwith (Dùor:0, andthematrix E*form > Iand'is,'it,...,'i*-1,'i*€
{1,...,n}tobe
(Ê^)n,oo...o^-r,
xa
--
(B^)¿'-¿!"¿-*-'¿^
(s.8.4)
_(Do)oo
The generating function of the offspring probability distribution can then be written
in matrix form
as
"f(")
:ã+ño"-rË Ê*s(^+').
(s.3.5)
m:l
8.3.2 Regularity and Mean Number of Branches
In order for the process to be regular (that is, non-explosive) [2] we require that
( ñ, for all 'i,i :
49
dt.¡
1,.
..
,fr.
Now, let (C*)ro...o^-ti*,¡ be the matrix that counts how many of the m
(8.3.6)
I
7 branches
emanating from a node are in phase 7 immediately after the creation of that node,
that
is'
T^.
(c^)oo,'n^-ti^,j:Ðt{ur: j}.
(8.3.7)
k:0
The expected number of branches in phase 7 given that the process began in
phase
i
can be calculated using [2]
M(t): exP(Aú),
where
A¡j:
(s'3's)
-(Ds)¡1b¿¡ and
_ 6¡,j
(8.3.e)
tm:1. B^C^)t)
(s.3.10)
s--e
In the
case of
the MT this is equal to
oo
M(t) :exp ((Ds +
The process is then
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
o subcritical if Àa (
o critical if
.
ÀA
:
0,
0, and
supercritical if À¿
where
194
)
)¿ is the dominant
0,
eigenvalue of ,4
8.3.3 Probability of Eventual Extinction
The final property we wish to discuss in this section is the probability of eventual
q. It
extinction, which we once again denote by
process theory [2]
is well known from branching
that if )A < 0 the process will eventually become extinct almost
surely and if À¿ > 0 then q < e component-wise.
It
is this final case that interests
us the most. Recall that the probability of eventual extinction of a continuous-time
Markovian multi-type branching process is the minimal non-negative solution [2] to
z(s)
:
g,
(8.3.11)
and for the MT t^c(s) can easily be shown to be
z(s) :
oo
d,+ Dos+ t
B^s(*+t),
(8.3.12)
m:I
using equation (8.3.5). We therefore have that q is the minimal non-negative solution
to
r¿(s) :
oo
d'-l Dos+ t
B^s(^+I):o.
(s.3.13)
m:L
We multiply this equation bV
s:
(-ro)-1
and re-arrange to obtain,
oo
(-Do) -1d +
| {-ar)-' B^s(^+')
,
(8.3.14)
m:1.
which is the form that is most useful for the discussion of the Depth and Order
algorithms to follow.
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
195
8.4 An Aside: Labelling the Nodes of an MT
Recall that in Chapter 3 we defined a node labelling system for binary trees. In
this system, each node was uniquely specified by a binary sequence. Suppose we are
at node
lrþl:
[0,ir,. ..,it), where 'h,...,'i¿ € {0,1}, and if this node is an internal
node, then
o the node that is at a depth of I + 2 and to the left of [r/] (which we call the
daughter node) is denoted by lrþ,0] : [0, 'it.,. . .,'d¿,0], and
o the node that is at a depth of I f 2 and to the right of [r/] (which we cali the
parental subnode) is denoted by lrþ,1] : [0, 'it,. . . ,i¿,7).
This node labelling system can by generalized to the Markovian tree. We begin
labelling the first non-root node of the MT bV [0]. Now, consider the node that has
1abel,
10,'i1,'i2,. . . ,i't]
where
'h,... ,i¿ àre non-negative integers, and suppose that mlL
branches emanate
from this node; m of these being the daughter branches and one of these being the
parental branch. We label the nodes at the tips of the unevolved daughter branches
AS
10,i,1,i,2,. . . ,it,O] [0,
il,
'i2,. . .
,'i¿,,1-]
10,'i1,i,2,. . . ,it,rn
- Il
and we label the tip of the unevolved parental branch, called the parental subnode,
by
f0, i,1,'i2, . .
.,'ü, m).
Let tþ denote a sequence of integers, such that the first is always a zero. The node
labelled Av lrþ] tras lTll indices. In addition, the number of indices of frl], lú1, Sirr".
the depth of that node. Let a(rþ) be the mapping that moves us up the tree from
lrþ]: [0,ir,...,,ir-r,i¿] to node, [0,ir,...,i;t].
We call
10,i.r,...,i;t] the parent
node of [ú]. W" define the root node of the tree to be ["(O)]. This labelling system
is depicted in Figure 8.4.1 for a Markovian tree.
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
196
0,0
0,1
0,0,0
1,0,0
1,0,2
1,1
0,0,1
0,1,1,2
I
0,0,1,0
1
0,0,2
0,
0,1,1,3
0,1,0,2,0
0,0,1,1
0,1
0,1,0
1,0,2,0,1
0,1,l,0
,0,2,I
0,1,
0,1,0,2,0,2
1,1,3,2
0,1,1,3,0
6, ,1,3,1
Figure 8.4.1: Labelling nodes in an MT
Just as in the MBT, that portion of a branch between the nodes [a(T/)] and
[ú] ir the ordered pair, ([r(ú)], [ú]). We write (t"(ú)l ,lrþ))(ù if this branch is an
internal branch. rffe write ([r(ú)],lrþl)@ if this branch is extinct, and finally we
write ([r(ú)],lrþD@ if this branch is unevolved. If a superscript is not specifled
then we just refer to that branch generically; its branch type is unimportant.
Now, suppose that there àte m
*
1 subtrees that emanate from node ltþl; m of
these being the daughter branches and one of these being the parental branch. rñ/e
represent the tree that commences from [c(T/)] by the ordered set,
{ ( t" (ú
)
], lrþDØ,
T[,þ,0), T[,þ,t], . .
.,
Tt
þ,*-
11,
Ty¡,,^1],
where T*,nís the topology of the 7-th daughter subtree that is based around lrþ,il,
for all
j:
0,7,,2,.
. . rrn
-
1 and Tþr,^l is the topology of the parental subtree based
around the parental subnode lrþ,*].
Let the set of branch points of an MT of topology
set of leaf nodes of an
MT of that
T
be denoted by
ßr,let
the
same topology be denoted by n r, we then have
8.
CHAPTER
THE GENERAL MARKOVIAN TREE
r97
that,
Nz:B7Un'7,
is the set of nodes of an MT of topology
7.
Since the number of daughter branches that are generated at each internal node,
[ú],
i. finite but unbounded,
we let,
o(rþ)
: max{j , [rþ, j] e Nz],
be the total number of branches that emanate from
(8.4.1)
[r/]. The parental branch
is
always that branch that is created from nodes, [r/] and lrþ,"(rþ)), that is
(l',þl,lrþ,
"(rþ)l)
Suppose that [r/] is either the root node or an internal node, then let the function,
I
be defined by,
e(o(o))
and for lrþl
I
[0],
l*Q)1,
0(rþ):
The function
:
I
lrþ,"(rþ)l
is well defined, and maps a node, lrþ] of depth
løl to the parental
that emanates from tþ andwhich is at a depth .f lrþ]-l1. Therefore
subnode, lrþ,
"(rþ)l
0r(lþ) traces the pathway of the parental branch that commences from node Ty',
provided that the parental branch is of at least length k from node [T/]. Clearly,
0o(rþ)
:
[*(rþ)],
[rþ]. Finally,
[T/])
if [T/] is a node, then, /(a(T/))
denotes the phase of the branch
immediately after la(rþ)1. The phase of the parental branch, (lrþ1,10(rþ)l)
immediately after the node [T/] is denoted by ór(rþ)
8.5 The Depth Algorithm
The Depth algorithm in the context of the MT is the continuous-time analogue of
the algorithm of Harris [9]. The Depth algorithm has a very interesting physical
CHAPTER
8,
THE GENE,RAL MARKOVIAN TREE
198
interpretation, similar to that in the MBT, namely, that the maximum depth the
Markovian trees can reach at each step of the algorithm can increase by one only.
Consequently, we follow a procedure that is similar to Chapter 7.
The Depth algorithm is the recursion on the following set of equations,
: (-Do)-'d
s(¿) : (-Do)-'d+ t
m:I
s(0)
(8.5.1)
oo
GDo)-'B^s(^+t) (l - 1), for, I
Definition 6 The depth, õ(T), of an MT of topology T
i,s defi,ned
)
to
1
(8.5.2)
be,
6(r): #tr"{l'i l}
The MT,
6q)
7
depicted in Figure 8.5.1 has,
##n {l'i l}
{ lol, lo, 1l, 10,21, 10,31, 10,1,01, 10,1,31,
10,
a,
rl, 10,3,1,11, 10,s,1,1,01 }
5
0,1
3,1
0,1,
3,I,I
0,1,3
0,3, 1, 1 ,0
Figure 8.5.1: An MT of depth
5
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
Lemma 17 lim¿-oo lf @l < oo. almost surely, if and onlg
199
i,f
limÞ*õ(T(t)) <
*,
almost surely.
Proof :
The proof is similar to Lemma 10 from Chapter
7
Since trees that eventually becomes extinct are of finite depth, almost surely, the
probability of eventual extinction of an MT is
q:
:
pllrwtl.-ld('(ú))l
Plõ6w) < ool ó("(,þ))l
(8.5.3)
Let q(l) be the probability that a ftee, T*, commencing with one branch will even-
tually become extinct under the taboo that 6(T*l)
< If
1, for all
I > 0.
That
is'
q(t):Pl6qwì<t+tlø('(ú))ì
(8.5.4)
: (-Do)-'d,
(8.5.5)
Notice that
q(0)
because an extinct tree of zero depth cannot undergo any observable transitions.
Theorem 18 The sequence {q(l)}, for I } 0, defined by equati,ons (8.5.1 and
(5.5.5) 'is monotoni,cally 'increas'ing and conuerges to the uector q. The sequence
{q(¿)} also sati,sfi,es,
s(0) :
"(¿)
Proof :
(-Do)-'d
(8.5.6)
oo
: (-Do)-td+t(-Do)-'B*s*+'(l-1),
once again the proof
J ri,,
l> 1.
(S.5.7)
theorem follows a very similar format to
Theorem 11 from Chapter 7. The fact that {q(¿)} is monotonically increasing is ob-
vious. That it converges to g is also obvious since lim¿-- q(t)
¿
+
1ló('(,Ð)l:
Plõ(Twr) <
-ló(.kþ))l:
:
lim¿-- Pl6ØWì <
q.
To show that q(l) satisfies equation (8.5.7) let us understand the physical evo-
lution of the process. There are only two pathways with which a tree of depth,
CHAPTER.8. THE GENERAL MARKOVIAN TREE
d(TWl
< I + 1, can eventually become extinct.
200
The first is a direct extinction,
where the parent branch undergoes a catastrophic transition before any births. The
probability of this scenario is just
(-ro) rd. In the second pathway the parent
undergoes an observable transition
at node
number of daughters with probability
[T/] spawning a finite
Clearly, in order for the
Ði:reDo)-'B^.
tree to eventually become extinct with depth ô
but unbounded
< l+1, all the daughter
subtrees and
the parental subtree must each independently become extinct with depths ô
The probability of this second pathway is given bv
< l.
Di:, ?Do)-tB*q*+I(l -
7).
Hence we have that,
oo
q(t): (-Do)-'d+ t ?Do)-'B^Q^t'(l m:7
(8.5.8)
1)
and the proof is complete.
At the l-th step of the algorithm the space of extinct trees that
includes all those trees from step I
plus all those trees of depth 6
-
are measured
1, that is, those trees that have depths, ô
< l,
: l. Consequently, at each step only a finite number
of extra trees are included. In Section 8.7 we discuss the Order algorithm which
includes infinitely many extra trees at each step and therefore converges at a faster
rate than the Depth algorithm but before we can do this we deflne the order of an
MT.
8.6 The Order of an MT: Definitron
Similarly to Section 7.4 we define the order of an MT. Now, let
number of internal nodes along the parental branch of
Definition 7 The order of a
topology TVrl, denoted bE
a
A,(0r(rþ))
k
"
@
si,ngle branch i,s
?,þ
))
:
¡
0.
Ng^l)
denote the
T*,
The order of a node, 0k(þ), of
i,s gi,uen by,
: r,r,.ïftã ¡¡,¡ _ r{a
(Tw
*
¡)},
1,¡¡,
a
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
207
for k:0,1,,2,...,N(TW.). Finally, the order of TWI i,s gi,uen
7
As in Chapter 7
it
by,
t r:o,rl1,lr1r,l {o(P*(''))}
is clear that we need to calculate the order of an MT recur-
sively. The procedure is similar to that described in that chapter. To illustrate the
procedure in the MT case we calculate the order of the topology depicted in Figure
T
8.6.1. The order of the tree,
depicted in the figure is,
T
[0,1]
0,2,2
[0,0,3]
ï
0,1
T¡o,r,r1
0,0,
0,2,2,7
1l
,01
ï0,0, 1,11
Tio, t,2l
10.0.1.21
Figure 8.6.1: An example of an order calculation
O(7)
:
1
+ max{O"(0), O,(0,2), O,(0, 2,2)}.
Now,
O,,,(0, 2.,2)
since
T¡o,z,z.o1
:
0(7¡6,2,2,01)
:
0,
consists of a single branch. Similarly,
O,,(0, 2)
:
max{O(7fo,r,ot), 0(7¡¡,2,11)}
:
0,
since both daughter subtrees, T¡o,r,o1 and T¡o,z,tl each consist of only a single branch.
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
202
Node [0] is slightly more complicated, because the daughter subtrees are more
complex than single branch trees, but still
O" (0)
:
max{O(T¡o,o1),O
(7¡s,r1 )
}
Now
O(7¡6,q)
:1*O'(0,0),
since there is only one branch point on the parental branch of Tp,ol. The order of
this branch point is,
O,, (0,
0)
:
max{O
(7¡o,o,o1
), O (7to,o, rl ), O (7¡¡,0,21 ) }.
The orders of 7to,o,ol undT¡o,o,rl are both zero,because these daughter subtrees each
consist of only one branch. On the other hand, T¡o,o,rl has order,
0(7¡6,0,11)
:
1
*
O'(0,
0, 1),
because T¡o,o,r7 has only one node along its parental branch. Now
O",(0,0, 1)
:
max{O(7¡o,o,r,ol), O(Tto,o,r,rl)}
:
0,
since both subtrees are only of single branches. Hence,
0(7¡6,0,11)
:
1
*O,,(0,0, 1) :
1
*o :
1,
and therefore,
O",(0,0)
so
:
max{O(7lo,o,ol), O(7to,o,tl),O(T¡s,o,E)}
max{1,0,0}
:
1,
that,
O(T¡r,q)
:
The order of the subtree Tp,rl
1.
:
1
+ O,,(0, 0) :
"un
1
l7 :
2.
be found in a similar manner and is O(7¡0,¡)
So,
O"(0)
:
max{O(T¡o,o1),O(7¡s,r1)}
:
max{2,1}
:2,
:
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
and as a result the order of
O(7) :
1
T
203
is,
+ max{O"(0), O,(0, 2), O,(0, 2,2)}
:
1 -l-
max{2, 0,0}
: 7 l2 :
3.
Trees that are topologically isomorphic can have different orders. Rotating the
nodes changes the parental branch and the daughter subtrees at each node along
the parental branch and this changes the calculation of the order.
We stated in the case of the MBT that the number of topologies that exist at any
order I
such
)
1 are infinite, and we saw this because a tree of order I can be constructed
that at each node ofthe parental branch the daughter subtree has order at most
I - L, and there can be any number of parental branch subnodes, so the possible
number of topologies is infinite. This generalises in a similar manner to the MT; at
each and every internal node of the parental branch
most I
-
1 then the order of the tree is
l.
if the order of the node is at
Parental branches with any number of
internal nodes can be created and so the number of topologies with order I is clearly
inflnite.
Lemma 19 lim¿-"o
Proof :
8.7
lfl .
æ
i,f and onty i,f
O(7)
(
@, almost surely
Once again the proof is similar to Lemma 14 from Chapter 7
The Order Algorithm
The Order algorithm, as we shall see, is a significant improvement on the Depth
algorithm. This algorithm reduces to the MBT Order algorithm developed in Chap-
ftr 7 if we restrict the process to spawn only one daughter branch at each branch
point. Let us re-write equation (8.3.14)
s:
as
oo
(-Do) -rd, +
u*("^ * ¡{r))s,
!{-lr)-'
m,:l
(8.7.1)
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
where 1(1) is the n x n identity matrix.
204
If we substitute this equation into the right
hand side we obtain,
oo
(-Do)-'d+
s
t
eno)-'e*(r^
ø ¡('))
?Do)-'d
rn:1,
where
X
+
(å,-^
:
(-Do)
-L
r',',)) (å,-"
)-'B^("-
d+
x (-
)-'
B^("- r r,',))
"
Do)-' d + x2 s
: Dn:r? Do)-t B^ ( 6 ¡(l) )
"nz
Now if we do this I times, we obtain,
(8.7.2)
":!xr?nù-'d+R¿(s¡,
k:o
where A¿(s)
: Xt*rs is the remainder term. Now if we take the limit as I --+ oo we
obtain,
":i k:0 xr(-oo)-1a+B(s),
(s.2.3)
where r?(s)
:lim¿-ooA¿(s). The above expression is well defined
because the left
hand side, s
1 e componentwise.
need to be zero,
However, in general the remainder term does not
but we can show that for q, the minimal non-negative solution of
equation (8.7.1), the remainder term rR(s) is zero orr physical grounds, as follows.
Substituting q into equation (8.7.3) we obtain,
q:
oo
t
xr?nù-'a+ nçq¡,
(8.7.4)
k--o
and we know that since g is the minimal non-negative solution of (8.7.1)
it
is also
the minimal non-negative solution to the above equation. Now we also know from
branching process theory that g is the probability measure of all the sample paths
that eventually have zero living particles. Due to the transience and regularity of
the ctMMTBP these sample paths are equivalent, almost surely, to the space of
extinct trees that have a finite number of extinct leaf branches. The first term of
equation (8.7.4) is the probability measure of all those extinct topologies with
a
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
205
finite number of branches. On the other hand the second term can be interpreted
as being
the probability measure of those topologies that have an infinite number of
leaf branches that eventually become extinct. Once again using the transience and
regularity of the process this space has probability measure zero and so R(q)
:
0
for q. As a result we consider only,
: i xr?nù rd,.
"
(8.2.b)
k:o
Let
oo
u : D?Do)-'B-(q- I 1(')),
m:I
and set
s:
Ç1,
(s.7.6)
the probability of eventual extinction, in equation (8.7.5). Now
q and U have very interesting
physical interpretations based on their respective
equations above. Let us first interpret the matrix
[/. Consider, the rn-th term from
the summation, that is,
(J*: (-ro)-' 8,.(q* I
1('))
(8.7.7)
This term gives the probability that there are Tn daughters spawned at
each of which generates a subtree
a
branch point
that eventually becomes extinct. For the purposes
of this expression the parental branch remains alive. We call such a structure a U^-
unit; if we do not specify how many daughter branches there
are we call the structure
a [/-unit. Figure 8.7.1 represents a [/a-unit. The parent branch gives birth to four
daughter branches. These four daughter branches generate subtrees that eventually
become extinct, whilst the arrow on the parent branch indicates that its evolution
has been suspended, that is,
it is an unevolved branch. In general, the suspension of
the evolution of the parental branch is made manifest in equation (8.7.7) by seeing
that the rn daughter branches are made extinct by the q^
term,,
whilst the evolution
of the parental branch is governed by the identity matrix, T(l), and hence does not
evolve. Allowing the parental branch to remain idle while its daughters all become
extinct is possible because of the independent evolution of each branch subsequent
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
206
Figure 8.7.1: An example of a U+-unit
to its birth. To evaluate the matrix [/ we sum over aII m, since there is no restriction
on the number of births.
To construct an extinct tree, we connect the parental subnode of the previous
[/-unit to the root node of a new U-unit. An extinct tree can only be of finite size,
almost surely, so only a finite number of U-units can be connected. Following the
connection of the final U-unit, the parent branch must undergo a catastrophe before
any other observable transition. As an example, Figure 8.7.2 depicts an extinct tree
u4
u"J
U1
Figure 8.7.2: An example of a tree with three [/¿-units.
that is constructed from a Ua-wiL a Lþ-unit and a [[-unit before flnal extinction.
The probability of obtaining a tree with this description is easily deduced to be,
CHAPTER
U
8.
THE GENERAL MARKOVIAN TREE
tht^- Do)-td.
207
More generally,
e Dù-|d,
uk
(8.7.8)
gives the probability of generating an extinct tree from
k [/-units. In other
words,
the above expression is the probability measure of the space of extinct trees that
are constructed from any combination of
k U-tnits, with the parent branch of the
k-th unit undergoing a catastrophe, with probability (-Do)-'d, to render the tree
extinct. Therefore,
îur'_
k:0
"''d'
is the probability measure of the space of all possible extinct trees.
To recap, the structural subunits that can be used to generate extinct trees are
the U-units. These units are connected using their parental branches as we have
shown above. Consider a node, lrþ] e
ßr,
we can represent a
[/-unit whose daughter
subtrees are spar'¡/ned at [ú], u. the following ordered set of branches and subtrees,
{([r(ú)],
[,.i])(n)
,Tl,þ,ol,Tl,þ,t1,...,T¡,¡,o1,t¡-'rl,(lrþ],[0(t/)])(")],
where the parental branch is denoted
¡V ([ú] ,10(1þ)l)(") to emphasizelhe fact that
this branch is unevolved.
Now, let B*l?k (rþ)] be the event that a parental branch which commences from
node [T/] has undergone
k
observable transitions since
daughter branches are spawned.
If
l,l]
such
that at 0k(þ) m
we do not specify the number of daughter
branches we write, ß[0r (rþ)) for the event that a parental branch which commences
from node [r/] has undergone k observable transitions since lþ1. Let Al0r(rþ)l be the
event that the parental branch has undergone
[T/]
k-
1 observable
transitions since node
followed by a catastrophic transition at node l0r(rþ)). As before, let þr(ekþ¡;))
the phase that the parental branch was in immediately after fhe
l7k
te
(1þ)]-th branch
point. The initial phase of the parent branch i" óo(rþ) The initial phase of a tree
of topology TWt is denoted AV ó@(rþ))
CHAPTER
8.
Definition 8
THE GENERAL MARKOVIAN TREE
The matrir
u
:
U
i,s defi,ned
208
to be
plsle@ù1, o"(p(,,r))) < oo &
ó,(0(rþ))10,þtì1,
(8.2.e)
for att fþ)
Definition 9 The uector q
i,s defined as,
e: P[o(rrú]) < ool4(a(,þ))1,
for alt
lrþl
I
*Q)
Expressed in this way, we see
node, [ú]
i"
(8.7.10)
U as being the probability that beginning at
some
phase ór(rþ), a branch point eventually occurs at node l0(rþ)1, the orders
of each of the daughter subtrees are finite, so, O,, (0(',þÐ) ( oo, and the parent
branch is suspended at node [á(T/)] in phase ór(0(rþ)). The matrix,
[/, is actually
independent of the position of the initial node because the subsequent evolution of
any branch that is spawned from that node is independent of the rest of the tree
immediately after its
birth. The probability of eventual extinction of the tree T¡,¡.,1,
g, is the probability tÌirat T*, has finite order as ú --+ oo, given that
it
commenced
from node l*þþ)l in phase, ó("(rþ)).
The Order algorithm to determine the minimal non-negative solution to equation
(8.7.5) is
"(0)
:
(-Do)-td,
(8.7.11)
oo
x(t) : !{-4.)-LB*(s^(t -
1) ø
l{r)¡,
(8.7.12)
m:l
oo
.s(¿) : Ð x* (t)(- Do)-'d,
k:o
(8.7.13)
forl>1
Theorem 2o The
u (t)
:
sequences
{U(t)l¿ >
t}
PlBl7 (1þ)1, a"(0(,þ)))
and {q(t)l¿
> o} defined by
. I k ó,(0(,þ))ló,þþ)l
(8.7.14)
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
209
and
q(t):
sati,sfE equati,ons
ela@¡,¡,) < ¿+
(8.7.11)-(5.7.13). The two sequences are monotoni,cally
and respectiuely conuerge to the matri,r
Proof
:
tló(.01ù)),
(s.7.15)
i,ncreasi,ng
U and the uector q.
We first show that the sequences {t/(¿)} and
{q(l)}
defined by (8.7.1a)
and (8.7.15) monotonically increase and converge to U and q respectively. T[ivially,
{U(l)}
and
{q(l)}
are monotonically increasing. F\rrther,
,r1*v(¿)
:
lim Plswrv¡| a"(0(,þ)))
:
plß10(1þ)1, a"(o(rþ)))
.-
.
¿
k
¿.
ó"(0(,þ))lo,røl
ó,(0(rþ))lþ,(,þ)ll
rr
u)
and,
lim P[o(ztú]) < ¿ + rld(,(ú))l
,t1i ø(t)
elaQ¡¡,) < *lø('(ú))l
q
The matrix, U^(l) is the probability that beginning at some node ltþ] and phase
ór(rþ), a branch point eventually occurs at node l0(rþ)1, generating rn daughter
subtrees such that the orders of each of the daughter subtrees is less than l, so,
O"(9(ti))) < l, and the parental branch is
suspended
at node [d(T/)] in
phase
ór(0(rþ))). Therefore we have,
u*(t)
:
:
:
<t k ó,(0(rþ))ló,(,þ)l
< t, k óo@þþ))lórþþ)l
*.roïuå_,1{o(zr,,r,*r)}
o(T¡.¡,ot) < ¿, .. ,Ø(Tw,-_il) < ¡, k ón(er*l)lórþþ)l
Plß*le?þ)1, o,(e?Ð)
plß*leþþ)1,
plß,"10(1þ)1,
where in the second step we have used
a"(0(lþ))
:
0,,ï%_,
{a(rvt,,ù}
CHAPTER
8,
2t0
THE GENERAL MARKOVIAN TREE
and in the third step we have used the fact that
(Tw,Ð,ù} <
is equivalent to A(T¡a,n1)
t,
',,ïH_,{o
< I for aII k : 0,1,.. .,m - I.
daughter subtree must be extinct with order less than
l.
Thus, each and every
Since each event occurs
independently we can write,
plß^10(1þ)1, o(7¡.¿,ol) <
u^(t)
¿, . .
.,a(Ttt,,-_11) < t
k
óp(o(rþ))ló"(þ)l
plftþl,leþÐD@, ó@(,þ,0)),.. ., ó(o(,þ,rn-r)),
t
t
ó(a(tþ'o) ó(a(tþ'm-7))
k
X
Pla(Tbþ,q)
.
¿lø("(,r/,0))]
óo,,þþ))rd"(r/)]
... Pla(rþþ,--l1) < tlô@(rþ,-
- 1))]
(8.7.16)
Now the second step can be understood when one notices that ß^10(rþ)l is the event
that
a branch
point occurs to make (lrþ],leþþ)]) an internal branch with nz daughter
branches, and the probability of this is
just (-Do)-tB-; the terms elA(f¡,¡,,n) <
4óþþ,k)] are just the probability that the subtree T¡,p,r1becomes extinct with order
at most l-7, inotherwords, q(l- 1), for allk:0,1,...)rn- 1. Wetherefore
have
that equation (8.7.16)
is
u^(t): (-Do)-'n,,(q^(t-
1)
81,',),
(s.7.17)
where the Kronecker product with 1(1) represents the fact that the parental branch
is frozen with probability 1. Now since a [/-unit may be constructed by any finite
number of daughter branches we sum over all possibilities and therefore we have
that,
u(t)
:Ëf -r.l-'n*(q^(t- 1) I 1,',)
m:I
(8.2.18)
Let Tfi, be a topology that is based around node [T/] whose parental branch has
undergone
k branch points before undergoing a catastrophic transition. The first
of these internal branch points
ir [ú].
The probabitity that this tree has order,
CHAPTER
8.
THE GENERAL MARKOVIAN TREE
Agú) < t+ 1 is given by, PlAQfti) < I +tl$(a(tþ))l
27r
and
it is not hard to see that
this is equivalent, to,
plaØtl <t+tló(a(1þ))l:
because at the
l+ tld("(ú))1,
PlAl7k(,Ðl,a!ú,t) <
k*1 parental subnode,l0r(rþ)],
(8.7.1e)
from [a(T/)], a catastrophic transition
occurs rendering the parental branch extinct. Now, recall that the order of a tree is
the maximum of the orders of all the nodes along the parental pathway, therefore,
we have that,
P lAlok þþ)1, ØQf¡,)
< t + tlo(,@ù))
(,þ))
P lAlïk ('r) 1,,:il_?,ä_,a
"@o
<
¿
|
:
d (CI (,/ ) ) I
.
However, the above expression is equivalent to each of the individual nodes having
order less than l, so
PlAllk?þ)1,,:il_1ä_, a,(00
PlAllkþÐ1,
o", (rt')
.
. tló('(l'))l
(ú))
t, an(o(rþ)) 1t,...,
o",
(0r-'(rþ)) < tlf(*(lù))
But since each of the nodes and their subtrees evoive independently, we can instead
write,
P lAllk þÐ1,
a(0' (,þ))
o:il_?,ã_,
t
óoþþ)
t
<
plß"r4¡_rlrþ1,
tló(a(1þ)
a,(rl)
)l
:
. t, a(rþ) k ór(rþ)10(*?1,))l
óp(ok-|(,þ))
xPlß"çs1,þ))-'ld(ú)]
k ór(0(rþ))lør(rlùl
o" (0r-'(rþ)) < t, Q(7k-'þþ)) k
, a"(o(rþ)) < t,
xPlß"çsr-'ç,¡,¡¡_ll0r-t(,r/)1,
a@(1þ))
ó, (er -' (,þl) | O, (er -, (,þ))l
xrl,\ek
where
O(/)
(r¡t)llór(0r-'(rÐ)1,
is just shorthand for ó(rþ,0),...,ó(rþ,"(rþ)
(s.7.20)
-
1), the phases of
each
daughter branch emanating from an internal node [T/]. The first k terms of equation
(8.7.20) are each
just the definition of U(l), and the last term is just equal to
CHAPTER
(-
8.
THE GENERAL MARKOVIAN TREE
212
Do)-'d, because after the k-th branch point of the parental branch,
a catastrophe
must occur. Hence we can write,
PlagÐ < ¿ + tld("(ú))l
:
uk
(D?Do)-'d.
(8.7.21)
However, to obtain q(l) we must sum over all the possible number of branch points
of the parental branch, so,
q(t)
: Ër
k:o
@(rÐ < t +rld(,(ú))l
oo
: Dur(t)(-Do)-'d.
k:o
The number of topologies that have order I
>
cause such trees can be constructed by combining
(8.7.22)
1 is infinite. We can see this be-
[/-units together, whose individual
orders must be less than I - 1. We can combine any number of these units together.
To re-construct the topologies of order I we can connect any number of [/-units from
the pool of units of orders I
- 7. At each iteration
of the Order algorithm, this
is
exactly what we do. Thus at each step, k, we recombine all the topologies of order
k- 1 and below to obtain
all the topologies of order k and below; there are an infinite
number of ways of combining the topologies of order less than
k.
Hence, at each step
we are including infinitely more topologies. In contrast, the Depth algorithm, which
is similar to the Harris algorithm in the discrete-time multi-type branching process,
only adds a finite number of new topologies at each step, in fact, if there were
topologies at the k-th step, an extra N"(k)
step, clearly a finite number for all frnite
-
¡ú(k)
*
1 are included at the
l/(k)
k+
1-th
k. As a result, the Order algorithm is a
much more efficient algorithm to determine the probability of eventual extinction of
a multi-type branching process.
The ease with which algorithms can be developed for the MBT carries across to
the MT. In this chapter we have developed the Depth and Order algorithms for the
MT. The Order algorithm is a novel way of calculating the extinction probability
CHAPTER
8,
THE GENERAL MARKOVIAN TREE
273
for a cIMMTBP, an algorithm which is a significant improvement on the algorithm
of Harris [9]. The MT representation of the ctMMTBP should provide the basis for
better algorithmic analysis of the cIMMTBP and as a result, allow it to play a more
prominent role in modelling physical phenomena.
Chapter
I
Conclusions and F\rrther Research
9.1
It
Conclusions
is widely believed that the rates of evolution, at the genetic level and at the
macroevolutionary level [6, 22],have not been constant throughout all time. At the
macroevolutionary level, these rate changes aiter the overall topologies of phyloge-
netic trees thus, changes in the speciation and extinction rates of species have the
effect of altering the imbalance of trees as they evolve. The simple models, such as
the crBD and PDA models, models that do not account for rate variation and have
cleariy been shown to be inadequate in generating topologies that agree with observation 1I1,22,30].
It
has become increasingly evident 17,221that more complex
models of macroevolution are required in order to aid in the inference of phylogenetic
tree topology. One step towards developing more sophisticated models is to allow for
rate variation. Consequently, an excellent candidate for a macroevolutionary model
is the ctMMTBP,17;22,26], but the distinct lack of adequate numerical methods
has hampered the use of the cIMMTBP as a modelling device.
In this thesis we
have addressed:
o the need for a reasonable macroevolutionary model based on the ctMMTBP,
and
2L4
CHAPTER
9.
2t5
CONCLUSIO¡\TS A¡úD FURTHER RESEARCH
o the need for some algorithmic approaches to the cIMMTBP.
We have developed a model of the macroevolutionary process based on the
binary-branch point cIMMTBP, which we have called the Markovian binary tree.
To transform from the cIMMTBP branching structure to the MBT transition struc-
ture, singular branch points (cIMMTBP) were interpreted as hidden transitions
(MBT) and binary branch points (ctMMTBP) were interpreted
as being observable
transitions. Observable transitions were regarded as corresponding to nodes in the
phylogenetic trees, while hidden ones were not.
With this subtle change of interpretation,
we were able
to describe the dynamics
of each branch of an MBT with a Markovian arrival process (MAP). As a result, the
time until an observable event or to the extinction of a branch need not be exponentially distributed and there exist correlations between the offspring distribution
and the lifetime of a branch. Using this MAP interpretation and the fact that the
tree topologies are binary we represented the cIMMTBP as a level-dependent quasi-
birth-death process. The states of the process were given by the number of branches
and the phase that each branch was
in. This representation allowed
us to unam-
biguously keep track of the daughter and parental branches and thus reconstruct
tree topologies.
In Chapter
5 we
demonstrated that the MBT subsumed all of the simple macroevo-
lutionary models such as the crBD, the PDA and the sPDA. What was of most
signifi.cance however, was that we showed that the multi-rate (MR) model of Pinelis
[26] was also subsumed by the
MBT. The multirate model (MR) is also based on the
binary-branch point cIMMTBP. In the MR model however, Pinelis [26] introduced
the concept of a quasi-stable branch, that is, a branch that cannot ever become
extinct or give birth to any daughter branches. The inclusion of this branch type
and the necessary pruning of extinct branches from the tree topologies had the effect
of complicating any analysis performed with the MR model. In fact, even showing
that the MR subsumes the simple models of macroevolution is not a trivial task. We
were abie to prove in Chapter 5 however, that the MBT model did indeed subsume
CHAPTER
9.
CONCLUSIO¡\IS A¡üD FURTHER RESEARCH
276
the MR model and that there was no need for quasi-stable branches and the pruning
of extinct branches could in fact be discarded. Thus the MBT not only subsumes
the MR but is also amenable to simpler analysis.
Having demonstrated that the MBT is the most general model, we wished to test
it in a macroevolutionary environment. As stated
on numerous occasions during this
thesis, the level of imbalance of phylogenetic trees is important because
it
has the
potential to show how the rates of evolution have changed over time. Therefore
any reasonable model of the macroevolutionary process must have the flexibility to
account for these imbalances. The imbalance algorithm was developed in Chapter
6.
The imbalance algorithm determined the mean imbalance of an MBT model
conditional on tree size. This algorithm therefore,
1. extended the work of Heard
[tt] and Rogers
[aO]
who did similar things for
the crBD and PDA models, and
2. showed that the MBT is suffi.ciently flexible that any mean imbalance is possible conditional on tree size!
The combination of flexibility and algorithmic tractability make the MBT a formidable
model in the macroevolutionary domain.
The subtle change of the interpretation of the branching structure of the ct-
MMTBP to that of the MBT ailowed us to develop two further algorithms in
Chapter
7.
These algorithms determine the probability of eventual extinction of
the process in the interesting super-critical domain. The algorithms that we
veloped
to solve for the probability of eventual extinction
de-
were called the Depth
and Order algorithms. Surprisingly, both these algorithms did not require use of
the level-dependent QBD representation of the MBT. This can be attributed to the
independent evolution of each branch of an MBT. Further, we demonstrated that
the Order algorithm was more efficient than the level-dependent logarithmic reduc-
tion algorithm. F\rrthermore, it was also demonstrated that the sample paths of the
CHAPTER
9.
CONCLUSIO¡\IS A¡úD FURTHER RESEARCH
277
Neuts algorithm at each step could be transformed to the set of the binary trees
measured at the equivalent step of the Depth algorithm.
The ease with which the algorithms could be developed and implemented in
the special case of the binary-branch point cIMMTBP was transferred to the general cIMMTBP. That is, we reinterpreted the transition branching structure of the
general cIMMTBP
in exactly the same way as we did in the MBT case: singular
transitions corresponded to hidden transitions, and non-singular transitions corresponded to observable transitions. Using this interpretation we once again developed
the alternative representation of the general cIMMTBP which we called the general
Markovian tree (MT) in Chapter 8. It was then relatively straightforward to develop
the Depth and Order algorithms for the MT in a similar fashion to the corresponding
algorithms in the MBT environment.
9.2
Future \ /ork
The essential groundwork for the MBT as a macroevolutionary model has been laid
out here.
It
possesses
tion. It is flexible,
was suffi.cient
it
all the attributes that make
as witnessed
it
a good model for macroevolu-
by the fact that a one parameter four phase model
to span the entire range of mean imbalances for trees of size 5 and
is readily analyzable as witnessed by the fact that we developed a number of
algorithms that were easy to implement.
What remains to be done is to appiy the MBT to macroevolution. More specifically, to determine how the MBT can be fitted to the data so as to generate an actual
model. Once this has been achieved, suitable tests of the models performance need
to be devised and implemented. The statistically fitted MBT model can be applied
to well known phylogenies to
see
what results are obtained. In other words, what
topologies are generated by the MBT with highest likelihood? How do they differ
from those of other studies? Does the MBT predict topologies that more closely
represent the true topology?
CHAPTER
9.
CONCLUSIO¡\IS A¡\ID FURTHER RESEARCH
278
Finally, representing the ctMMTBP as an MT gives to the cIMMTBP a richer
transition structure. A transition structure that has correlations between branch
lifetime and branch offspring distributions.
It
is our belief that since the MT
is
more amenable to algorithmic analysis, the MT representation may serve as a good
starting point to develop other algorithms for the ctMMTBP. Algorithms that will
enable the cIMMTBP to be used in a wide variety of modelling contexts.
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